IN  MEMORIAM 
FLORIAN  CAJORI 


PLANE 
TRIGONOMETRY 


BY 


ARTHUR   GRAHAM  HALL,  Ph.D.  (Leipzig) 


Pbofessor  of  Mathematics 
University  of  Michigan 


FRED   GOODRICH  FRINK,   M.S.  (Chicago) 

Professor  of  Railway  Engineering 
University  of  Oregon 


NEW  YORK 
HENRY  HOLT  AND   COMPANY 


Copyright,  1W9, 

BY 

HENRY  HOLT  AND  COMPANY. 


CAJORI 


NarfajDoli  \$xt9S 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

This  book  has  had  its  origin  in  the  desire  of  the  authors  to 
meet  the  mutual  demands  of  mathematicians  and  engineers  for 
a  treatment  that  shall  more  completely  supply  the  needs  of  the 
technological  student.  It  is  believed  that  this  has  been  done  by 
enriching  the  subject  with  applications  to  physics  and  engineering, 
in  such  a  way  as  to  increase  its  value  at  the  same  time  to  the 
general  student.  The  present  volume  is,  moreover,  based  upon 
a  preliminary  edition  actually  used  for  several  terms  in  the  class- 
room. 

In  view  of  the  peculiar  situation  of  trigonometry  in  the  cur- 
riculum, the  course  has  been  kept  of  the  usual  length.  The 
topics  have  been  arranged,  however,  in  the  order  of  increasing 
difficulty,  by  postponing  the  more  abstract  but  no  less  essential 
study  of  the  functions  of  the  general  angle,  until  after  the  arith- 
metical solution  of  triangles.  The  abundance  of  exercises  and 
problems  will  give  the  teacher  large  opportunity  for  selection. 

The  discussion  of  the  slide  rule  is  inserted  because  of  the 
increasing  employment  of  this  useful  instrument. 

The   authors   gratefully   acknowledge    their    indebtedness  to 

Professor    E.    J.    Townsend   and   Professor   H.  L.  Rietz,   of   the 

University  of  Illinois,  and  Professor  A.  Ziwet  and  Professor  J.  L. 

Markley,  of  the  University  of  Michigan,  for  valuable  criticisms 

and  suggestions. 

ARTHUR  G.   HALL. 

FRED   G.   FRINK. 

Ann  Arbor,  January,  1909. 


iii 


CONTENTS 

PAET    I 
PLANE   TBIQONOMETBY 

CHAPTER   I 
GEOMETRIC   NOTIONS 

ARTICLE  PAOS 

1.  General  statement 1 

2.  Directed  line  segments 1 

3.  Positive  and  negative  angles 2 

Exercise  I        .         .         .         .    ^ 3 

4.  Rectangular  coordinates 3 

Exercise  II 6 

CHAPTER   II 
THE   ACUTE  ANGLE 

6.    The  purpose  of  trigonometry 6 

6.  Definitions  of  the  trigonometric  functions .  6 

7.  Relations  between  the  ratios 7 

8.  Signs  and  limitations  in  value 7 

Exercise  III 8 

9.  Fundamental  relations 9 

Exercise  IV 10 

10.  Functions  of  complementary  angles 11 

11.  Functions  of  45°,  30°,  60° 12 

12.  Functions  of  0°  and  90° •.         .         .         .12 

Exercise  V 13 

13.  Variation  of  the  trigonometric  functions  as  the  angle  varies       ...  14 

14.  Inverse  trigonometric  functions 15 

Exercise  VI 16 

16.    Orthogonal  projection .•        .         .  16 

Exercise  VII 18 

CHAPTER   III 
RIGHT  TRIANGLES 

16.  Laws  for  solution 19 

17.  Area  of  right  triangles .  20 


VI 


CONTENTS 


ARTICLE  PAGK 

18.  Method  of  solution 20 

19.  Trigonometric  tables 22 

20.  Errors  and  checks 23 

Exercise  VIII 25 


21. 
22. 
23. 
24. 
25. 


27. 
28. 
29. 

30. 

31. 


CHAPTER  IV 

LOGARITHMS 

Definition  of  a  logarithm    .        .        . 29 

Laws  of  combination ,        .        .         .30 

Common  logarithms 31 

Characteristic      ............       32 

Mantissa 33 

Exercise  IX .       34 

Interpolation 1         ...       35 

Numbers  from  logarithms 35 

Cologarithms .         .36 

Logarithms  of  trigonometric  functions 36 

Exercise  X       . 37 

The  slide  rule 39 

Exercise  XI .41 

Right  triangles  solved  by  logarithms .41 

Exercise  XII 44 


CHAPTER   V 


THE  OBTUSE  ANGLE 

32.  Definitions  of  the  trigonometric  functions  of  obtuse  angle 

33.  Signs  and  limitations  in  value 

34.  Fundamental  relations 

35.  Variation 

36.  Functions  of  180° 

37.  Functions  of  supplementary  angles 

38.  Functions  of  (90°  -fa) 

Exercise  XIII 


47 
47 
48 
48 
48 
48 
49 
50 


CHAPTER   VI 
OBLIQUE  TRIANGLES 

39.  Formulas  for  solution 52 

40.  Law  of  projections 53 

41.  Law  of  sines        ............  63 

42.  Law  of  cosines 53 

43.  Law  of  tangents 64 

44.  Angles  in  terms  of  sides 65 


CONTENTS  vii 

ARTICLE  PAGE 

45.  Area  of  oblique  triangles    ....,,....  55 

46.  Numerical  solution       .         .  ' 56 

47.  Case  I.     Two  angles  and  one  side 57 

48.  Case  II.     Two  sides  and  an  opposite  angle 58 

49.  Case  III.     Two  sides  and  the  included  angle 61 

50.  Case  IV.     Three  sides 61 

51.  Composition  and  resolution  of  forces.     Equilibrium  .         .         .         .         .  63 

Exercise  XIV 66 

CHAPTER   VII 
THE   GENERAL  ANGLE 

52.  General  definition  of  an  angle 70 

53.  Axes,  quadrants,  etc.   '       .         .         .         .         .        .         .        .         .         .71 

54.  Definitions  of  the  trigonometric  functions  .......  71 

55.  Signs  and  limitations  in  value 72 

Exercise  XV 73 

56.  Variation  of  the  trigonometric  functions    .......  74 

57.  Graphs  of  the  trigonometric  functions 75 

58.  Functions  of  270°  and  360° 78 

Exercise  XVI 79 

69.    Fundamental  relations 79 

60.  Line  representations  of  the  trigonometric  functions  .         .         .        .         .  80 

Exercise  XVII 82 

61.  Periodicity  of  the  trigonometric  functions 83 

62.  Functions  of  (k  •  ^±  a) ,         .        .        .83 

Exercise  XVIII 87 


CHAPTER  VIII 
FUNCTIONS  OF  TWO  ANGLES 

63.  Formulas  for  sin  (a  +  /3)  and  cos  (a  +  /S)    .         .         =         .        .        .         .89 

64.  Extension  of  the  addition  formulas 90 

Exercise  XIX 91 

65.  Subtraction  formulas 91 

66.  Formulas  for  tan  (a  ±/3),  cot  (a  ±  (S)  .         .         .         .         .         .         .92 

Exercise  XX    ....  92 

67.  Functions  of  twice  an  angle 93 

68.  Functions  of  half  an  angle  .         .         .         ,         .         .         .         .         .93 

Exercise  XXI  ....         .         .         .         .         .         .         .         .94 

69.  Conversion  formulas  for  products 95 

Exercise  XXII 96 

70.  Conversion  formulas  for  sums  and  differences 97 

71.  Multiple  angles 97 

Exercise  XXIII 97 


viii  CONTENTS 


CHAPTER  IX 


ANALYTIC  TRIGONOMETRY 

AETIOLE  PAGE 

6  0 

72.  Limits  of  - — t,  and  - — -,  as  6  approaches  zero 99 

sin  d  tan  0' 

Examples 101 

73.  De  Moivre's  theorem 101 

Examples 103 

74.  Graphical  representation  of  complex  numbers 103 

Examples 108 

75.  Exponential  values  of  the  trigonometric  functions 109 

Examples 110 

76.  Hyperbolic  functions 110 

77.  Exponential  and  trigonometric  series Ill 

Examples 115 

78.  Computation  of  trigonometric  tables  ........  116 

79.  Proportional  parts 116 

80.  General  inverse  functions   .         .         .         .         ,         .         ,         .         .         .117 

81.  Logarithmic  values  of  inverse  functions 118 

Examples 119 

Review  Exercises '     .         .  120 

Formulas 130 

Answers 137 

Index         , .  147 


TRIGONOMETRIC   AND   LOGARITHMIC   TABLES 

I.  Common  logarithms  of  numbers 3 

II.  Logarithms  of  the  trigonometric  functions        .         .         .         .         .         .25 

III.  Natural  trigonometric  functions        ........       71 

IV.  Squares  and  square  roots .....         o         ....       91 


TRIGONOMETRY 


GREEK   ALPHABET 


Letters 

Names 

Letters 

Names 

Letters 

Names 

A  o 

Alpha      ' 

I     I 

Iota 

P  P 

Rho 

B^ 

Beta 

K/c 

Kappa 

S(r  s 

Sigma 

Ty 

Gamma 

AX 

Lambda 

Tt 

Tau 

A5 

Delta 

Mm 

Mu 

Tu 

Upsilon 

Ee 

Epsilon 

N  p 

Nu 

$0 

Phi 

Zf 

Zeta 

S^ 

Xi 

Xx 

Chi 

H^ 

Eta 

Oo 

Omicron 

^,/. 

Psi 

ee 

Theta 

Htt 

Pi 

0  w 

Omega 

TRIGONOMETRY 

PART   I 
PLANE  TRIGONOMETRY 

CHAPTER   I 

GEOMETRIC  NOTIONS 

1.  General  statement.  It  is  assumed  that  the  student  is  well 
versed  in  those  theorems  of  elementary  geometry  concerning 
angles,  arcs,  and  triangles.  It  is  particularly  desirable  that  he 
be  familiar  with  the  measurement  of  angles  and  with  the  proper- 
ties of  similar  triangles. 

While  the  review  thus  suggested  is  left  to  the  student,  certain 
more  advanced  geometric  ideas  are  treated  in  the  remaining  arti- 
cles of  this  chapter. 

Throughout  the  course  the  student  should  make  continual, 
careful,  and  intelligent  use  of  such  drawing  instruments  as  are 
included  in  the  equipment  at  technical  schools.  In  case  such  sets 
are  not  available,  as  in  more  general  classes,  there  should  be  pro- 
vided at  least  a  straightedge,  with  graduated  scale,  a  protractor, 
and  a  pair  of  compasses  or  dividers. 

2.  Directed  line  segments.  A  point  which  moves  from  one 
position  to  a  second,  without  changing  its  direction  of  motion, 
traces  a  directed  line  segment.  Directed  line  segments  are  always 
read  with  regard  to  their  direction,  from  the  initial  extremity  to 
the  terminal  extremity. 

Two  line  segments  are  equal  if  they  have  the  same  length  and 
direction,  whether  their  lines  are  coincident  or  parallel.  Either 
of  two  line  segments  having  the  same  length  but  opposite  direc- 
tions is  said  to  be  the  negative  of  the  other.  If  one  direction  is 
taken  as  positive,  the  opposite  direction  is  negative. 

1 


GEOMETRIC   NOTIONS 


iXiius,  An  FigCli,. 


E  F         K  H 


HK^  BA  =  -  AB, 


Fio.i.  CD='2BA  =  -2AB. 

If  ^  is  the  initial  point  and  B  the  terminal  point,  the  line 
segment  is  read  AB,  and  in  this  notation  the  positive  or  negative 
direction  of  the  segment  is  expressed  without  the  aid  of  a  prefixed 
+  or  — .  In  case  the  segment  is  represented  by  a  single  symbol, 
as  the  letter  «,  the  direction  must  be  indicated  in  some  further 
manner,  as  by  a  prefixed  +  or  — ,  or  by  an  arrowhead  in  the 
figure. 

Two  line  segments  lying  in  the  same  line  are  added  by 
placing  the  initial  point  of  the  second  upon  the  terminal  point 
of  the  first,  each  retaining  its  proper  direction.  The  sum  is  the 
segment  extending  from  the  initial  point  of  the  first  to  the  termi- 
nal point  of  the  second.  Subtraction  is  performed  by  reversing 
the  direction  of  the  subtrahend  and  adding.  Line  segments 
having  the  same  or  opposite  directions  may  all  be  transferred 
to  a  common  line.  Their  addition  and  subtraction  thus  cor- 
respond exactly  to  the  algebraic  addition  and  subtraction  of  posi- 
tive and  negative  numbers. 

If  A,  B^  0  denote  three  points  arranged  in  any  order  along 
a  straight  line,  then 

AB^BQ=AQ, 

and  AB^BC^CA  =  ^. 

3.  Positive  and  negative  angles.  If  a  line  rotates  (in  a  plane) 
about  one  of  its  points,  an  angle  is  generated,  of  which  the  origi- 
nal position  of  the  line  is  the  initial  side  and  the  final  position  the 
terminal  side.  A  distinction  may  be  made  according  as  the  rota- 
tion is  clockwise  or  counter-clockwise  about  the  vertex.  The 
counter-clockwise  direction  is  chosen  as  positive.  Angles  are 
always  read  with  regard  to  their  direction  of  rotation ;  thus  if 
OA  is  the  initial  side  and  OB  the  terminal  side,  tlie  angle  is  read 
AOB.  This  notation  includes  the  direction  or  sign  of  the  angle, 
and  the  -f-  or  —  should  not  be  prefixed.  In  case  the  angle  is 
represented  by  a  single  symbol,  as  by  the  Greek  letter  a,  the 
direction  must  be  indicated  in  some  further  manner,  as  by  a  pre- 
fixed -f  or  — ,  or  by  a  curved  arrow  in  the  figure. 


POSITIVE   AND  NEGATIVE   ANGLES  3 

Just  as  with  line  segments,  reversing  the  direction  multiplies 
the  angle  by  -  1 ;  thus  BOA  =  -AOB. 

Two  angles  are  added  by  placing  them  in  the  same  plane  with 
a  common  vertex,  the  initial  side  of  the  second  coincident  with 
the  terminal  side  of  the  first,  each  retaining  its  own  direction. 
The  sum  is  the  angle  from  the  initial  side  of  the  first  to 
the  terminal  side  of  the  second.  Subtraction 
is  performed  by  reversing  the  subtraliend  and 
adding. 

In  Fig.  2, 

AOB  +  BOO=AOC, 

AOO-BOC=AOB.  ^:;^^ 

EXERCISE    I 

Solve  the  following  problems  graphically  : 

1.  On  a  train  running  40  miles  an  hour,  a  man  walks  4  miles  an  hour. 
Find  the  speed  of  the  man  with  reference  to  the  ground,  (a)  if  he  walks 
toward  the  front ;  (6)  if  he  walks  toward  the  rear  of  the  train. 

2.  The  man's  speed  with  reference  to  the  ground  is  10  miles  an  hour. 
What  is  the  speed  of  the  train  (a)  if  he  is  walking  5  miles  an  hour  toward 
the  front;  (b)  if  lie  is  running  8  miles  an  hour  toward  the  rear? 

3.  On  June  1  the  price  of  corn  was  50  cents,  and  during  the  succeeding 
ten  days  it  fluctuated  as  follows :  rose  2  cts.,  rose  3,  fell  1,  fell  2,  fell  5,  fell  3, 
rose  2,  rose  2,  rose  3,  rose  1.     Find  the  price  on  June  11. 

4.  During  a  football  game  the  progress  of  the  ball  from  the  middle  of  the 
field  was  north  40  yards,  south  25,  south  5,  south  10,  south  30,  north  50,  north 
10,  north  20.     Find  the  resulting  position  of  the  ball. 

Combine  graphically,  using  a  protractor  : 

5.  45°  +  30° ;  90°  +  45° ;  40°  +  35°  +  50°. 

6.  60°  -  45° ;  90°  -  50° ;  180°  -  120°. 

7.  30°  +  80°  +  55° ;  40°  +  60°  -  30° ;  60°  -  20°  +  70°  -  90°. 

8.  40°  -  70°  +  15°;  65°  -f  15°  -  90°;  75°  -  180°. 

4.  Rectangular  coordinates.  If  two  mutually  perpendicular 
straight  lines  are  chosen,  and  a  positive  direction  on  each,  the 
position  of  any  point  in  their  plane  is  determined  by  giving  its 
perpendicular  distances  from  these  fixed  lines.  The  two  lines  are 
called  the  axes  of  coordinates,  and  are  usually  taken  so  that  one 


GEOMETRIC   NOTIONS 


r^^ 


O^^-x. 


-^x 


^A 


is  horizontal  and  the  other  vertical.  The  point  of  intersection  of 
the  axes  is  called  the  origin.  The  two  determining  data  for  any 
point  are  called  its  coordinates.  The  horizontal  distance  from  the, 
axis  Oy  to  the  point  is  the  abscissa  of  the  point,  and  the  vertical 
.y  distance  from  the  axis   OX 

to  the  point  is  the  ordinate 
of    the    point.       The    point 
^^-jP^  C^^  whose     abscissa     is     x     and 
J^  ordinate    y    is    denoted    by 

the  notation  (a:,  ^).  Be- 
cause it  is  convenient  to  lay 
off  the  abscissa  of  a  point 
upon  the  axis  OX  and  the 
ordinate  upon  the  axis  OY^ 
these  axes  are  referred  to 
^^^-  ^-  as  the  axes  of  abscissas  and 

ordinates  respectively.  When  x  denotes  the  abscissa  and  y  the. 
ordinate  of  the  point,  the  axes  may  be  referred  to  as  the  X-axis 
and  the  iF-axis  respectively. 

The  distance  from  the  origin  to  the  point  is  called  the  radius 
vector  of  the  point.  It  is  known  whenever  the  abscissa  and  the 
ordinate  are  given,  since  the  three  form,  respectively,  the  hypote- 
nuse, base,  and  altitude  of  a  right  triangle. 

The  abscissa  of  a  point  should  always  be  read  from  the  F-axis 
to  the  point.  The  direction  from  left  to  right  is  chosen  as  posi- 
tive. Therefore  all  points  at  the  right  of  the  y"-axis  have  positive 
abscissas,  and  all  points  at  the  left,  negative  abscissas.  The  ordi- 
nate of  a  point  is  always  read  from  the  JT-axis  to  the  point.  The 
upward  direction  is  chosen  as  positive.  Hence  all  points  above 
the  X-axis  have  positive  ordinates,  and  all  points  below,  negative 
ordinates.  The  radius  vector  is  always  read  from  the  origin  to 
the  point,  and  is  always  considered  positive. 

It  will  be  noticed  that  the  abscissa  and  the  ordinate  are  equal 
to  the  projections  of  the  radius  vector  on  the  X-axis  and  P"-axis, 
respectively;  these  projections  will  henceforth  be  used  inter- 
changeably for  the  coordinates  themselves.* 

*  The  foot  of  the  perpendicular  dropped  from  a  point  upon  a  given  line  is  said  to 
be  the  orthogonal  or  orthographic  projection  of  the  point  on  the  line.  The  projec- 
tion of  a  line  segment  on  a  given  line  is  the  segment  from  the  projection  of  the  ini- 
tial point  of  the  given  segment  to  that  of  the  terminal  point.  This  kind  of  projec- 
tion will  be  used  exclusively  throughout  this  book,  unless  otherwise  expressly 
stated. 


RECTANGULAR   COORDINATES 


The  two  axes  divide  the  whole  plane  into  four  portions,  known 
as  the  first,  second,  third,  and  fourth  quadrants,  beginning  with 
the  upper  right-hand  quadrant  and 
numbering  counter-clockwise  about 
the  origin. 

If  two  points,  P  and  Q,  lie  in  a 
line  through  the  origin,  their  coordi- 
nates, with  the  radii  vectores,  form 
two  similar  triangles.  If  the  abscissa, 
ordinate,  and  radius  vector  of  P  are 
X,  y,  V,  respectively,  those  of  Q  are 
Jcx,  ky^  kv.  Fig.  4. 

EXERCISE  II 

1.  Plot  the  points  (2,  3),  (-3,  5),  (-2,  -4),  (1,  -3),  (3,  0),  (0,  4), 
(-5,0),  (0,  -2),  (0,0). 

2.  Plot  the  points  (3,  2),  (6,  4),  (12,  8). 

3.  Plot  the  points  (0,  5),  (4,  3),  (-3,  4),  (0,  -5). 

4.  Find  both  graphically  and  by  computation  the  radius  vector  of  each 
point  in  examples  1,  2,  3.     In  what  quadrant  does  each  point  lie  ? 

5.  Describe  the  location  of  all  points  whose  abscissas  equal  3 ;  whose  ordi- 
nates  equal  5  ;  whose  radii  vectores  equal  6. 

6.  Write  the  coordinates  of  the  vertices  of  a  square  of  side  a,  if  the  axes 
of  coordinates  are  taken  along  two  sides ;  along  its  diagonals. 

7.  The  hour  hand  of  a  clock  is  5  inches  long.  Find  the  coordinates  of  its 
tip  referred  to  the  horizontal  and  vertical  diameters  of  its  face  at  three  o'clock  ; 
at  six ;  at  eight ;  at  half-past  ten. 

8.  Compare  the  location  of  the  points  (2,  3),  (3,  2),  (  —  2,  3),  (  —  2,  —3), 
(3,  —2).     Describe  the  directions  of  their  radii  vectores. 

9.  Find  the  distance  from  (2,  5)  to  (6,  9)  ;  from  (-3,  2)  to  (4,  5). 

10.  Find  the  direction  of  the  line  joining  (6,  1)  to  (8,  3) ;  (4,  1)  to  (1,  4)  ; 
(6,  3)  to  (1,3);  (-2,4)  to  (1,1). 

11.  A  man  starts  from  the  southwest  corner  of  a  government  township  and 
goes  in  turn  to  the  northwest  corner  of  section  36 ;  northwest  corner  section 
22 ;  southeast  corner  section  3 ;  northeast  corner  section  5 ;  thence  to  the  point 
of  beginning.  Show  by  sketch  the  shortest  route  along  section  lines,  and  com- 
pute the  cross-country  distances. 

12.  A  city  is  laid  out  in  checker-board  fashion.  The  streets  are  eight  to 
the  mile  and  look  to  the  cardinal  points  of  the  compass.  It  is  proposed  to  in- 
troduce two  diagonal  (45"")  streets  extending  through  the  busiest  corner  to  the 
outskirts.  What  distances  will  be  saved  thereby  in  driving  from  this  corner  to 
each  of  the  points  specified  below?  Nine  blocks  east  and  six  blocks  north; 
5  W.  and  7  S. :  10  W.  and  10  N. :  1  E.  and  14  S. 


CHAPTER   II 


THE  ACUTE  ANGLE 


5.  Purpose  of  trigonometry.  One  of  the  principal  objects  of 
trigonometry  is  the  computation  of  triangles.  We  have  learned 
from  elementary  geometry  that  a  triangle  is  determined  when  we 
know  any  three  of  its  parts  (sides  and  angles),  at  least  one  of  them 
being  a  side.  These  data  enable  us  to  construct  the  triangle  ;  but 
elementary  geometry  does  not  teach  us  how  to  calculate  the  re- 
maining parts.  The  reason  is  that  sides  and  angles  are  expressed 
in  different  units.  It  is,  therefore,  desirable  to  measure  angles  not 
only  in  degrees,  but  also  by  means  of  lines,  or  rather  by  means  of 
ratios  of  lines.  This  can  be  done  as  shown  in  the  following 
articles. 

6.  Definitions  of  the  trigonometric  functions..  Suppose  the 
acute  angle  A  OB  (  =  «)  to  be  placed  on  a  system  of  axes  of  coor- 
dinates with  its  vertex  at  the  origin  and  its 
initial  side  OA  extending  along  the  X-axis 
toward  the  right.  Its  terminal  side  OB  will 
fall  in  the  first  quadrant.      (See  Fig.  5.) 

Any  point  P  in  its  terminal  side  possesses 

one    and   only  one    set  of    related   distances 

(two  coordinates  and  radius  vector). 

1^    > X  Its    abscissa    x(^=OM)^  its    ordinate 

^  (=zMP),     and     its     radius    vector 

V  (=  OF)    are    all    positive  and   con- 


tY 


O 


M 
Fig.  5. 


nected  by  the  relation         o  .     o       o 

•^  TT  -\-  y^  —  v^. 

The  six  ratios  between  these  three  distances  are  of  frequent 
occurrence  and  prime  importance.  They  serve,  indeed,  the  pur- 
pose mentioned  in  Art.  5,  and  are  given  the  following  names  and 
accompanying  abbreviations : 


^  =  sine  of  a  =  sin  a, 


=  cosine  of  a  =  cos  a, 
=  tangent  of  a  =  tan  a, 


=  cosecant  of  a  =  esc  a, 
=  secant  of  a  =  sec  a, 
=  cotangent  of  a  =  cot  a. 


FUNCTIONAL   RELATION 


7.  Relations  between  the  ratios  and  the  angle.  The  three  re- 
lated distances  of  any  other  point  P^  in  the  terminal  side  of  the 
angle  a  are  connected  with  the  determining  distances  of  P  by  the 
relation  ,        ,        , 

X       y      V 

It  follows  that  the  values  of  the  six  ratios  defined  in  Art.  6  do 
not  depend  at  all  upon  the  particular  choice  of  the  point  in  the 
terminal  side  of  the  angle,  but  are  determined  solely  and  definitely 
by  the  size  of  the  angle.  A  number  that  is  determined  in  value 
by  the  value  of  some  other  number  is  said  to  be  a  function  of  that 
other  number.  The  six  ratios  are  therefore  called  trigonometric 
functions  of  the  angle. 

This  relation  between  the  ratios  and  the  angle  is,  moreover,  a 
mutual  one,  such  that  a  knowledge  of  one  of  the  ratios  leads  to  a 
knowledge  of  the  angle.*  Thus  if  we  have  given  tan  a  =  J,  we 
can  construct  the  angle  a  as  follows  :  On  the  system  of  axes  OX 
and  OF  plot  the  point  P  whose  coordinates  are  (3,  2),  using  any 
convenient  scale.  Draw  the  line  OA  from  the  origin  through  the 
point  P ;  then  is  XOA  the  required  angle  a,  in  consequence  of  the 
definitions  of  Art.  6. 

It  appears  still  further  that  from  the  knowledge  of  any  one  of 
the  six  trigonometric  functions  the  remaining  five  can  be  found. 
Thus  in  the  foregoing  example  we  know  by  the  *  Pythagorean 
proposition  that  on  the  scale  employed  the  hypotenuse  or  radius 
vector  is  V9  H-  4  =  Vl3.     We  have  then  at  once 

sin  a 


cos  a 


Vl3 
3 


2 
tan„  =  -. 

Vl3 

seca  =  -— , 

cot  '^=  -, 

VT3 

CSC  «  =  —-—. 

V13 

The  properties  and  relations  of  these  functions  and  their  more 
immediate  applications  in  pure  and  applied  mathematics  constitute 
the  subject-matter  of  trigonometry.  The  science  takes  its  name 
from  its  origin  in  the  attempts  of  the  ancient  peoples  to  measure 
triangles. 

8.  Signs  and  limitations  in  value.  When  any  acute  angle  is 
placed  on  the  axes  of  coordinates  in  the  manner  prescribed  in 

*Tbis  statement,  as  well  as  some  others  in  the  present  chapter,  will  require 
some  modification  when  we  extend  the  consideration  to  angles  greater  than  90°. 


8  THE   ACUTE   ANGLE 

Art.  6,  its  terminal  side  will  always  fall  in  the  first  quadrant. 
The  abscissa  and  ordinate,  as  well  as  the  radius  vector,  of  every 
point  in  the  terminal  side  will  all  be  positive.  It  follows  that 
their  ratios,  comprising  the  six  trigonometric  functions  named  in 
Art.  6,  are  all  positive. 

In  no  case  can  the  abscissa  or  the  ordinate  of  a  point  be  greater 
than  the  radius  vector.  Indeed,  save  in  the  exceptional  cases  con- 
sidered in  Art.  12,  they  are  less  than  the  radius  vector.  Conse- 
quently, sin  a  and  cos  a  cannot  be  greater  than  unity  and  sec  a  and 
CSC  a  cannot  be  less  than  unity. 

EXERCISE  III 

By  careful  construction  and  measurement  find  the  approximate 
values  of  the  following  functions : 

1.   cos  60°.  2.   tan  30°.  3.    esc  45°. 

4.  cot  35°.  5.   sin  20°.  6.    sec  50°. 

Construct  each  of  the  following  angles  and  find  by  measure- 
ment the  values  of  all  its  functions,  given 

7.   sin  a  =  f .  8.   cos  a  =  y\.  9.   tan  a  =  ^\. 

10.  cot  a  =  y«3.  11.   sec  a  =  ^^.  12.   esc  a  =  f  i. 

13.   cos  a  =  ^f .  14.   sin  a  -  |^. 

15.  For  what  angle  is  the  tangent  equal  to  the  cotangent  ?  For  what  angle 
is  the  sine  equal  to  the  cosine  ? 

16.  Show  that  the  direct  functions  (sin  a,  tan  a,  sec  a)  are  greater  or  less 
than  the  corresponding  complementary  functions  (cos  a,  cot  a,  esc  a)  respec- 
tively, according  as  the  angle  a  is  greater  or  less  than  45°. 

17.  Can  sin  a  and  tana  be  equal?    When  do  they  approach  equality? 

18.  Show  that  tan  a  •  cot  a  does  not  depend  on  a.  Show  that  the  same  is 
true  of  sin  a  •  esc  a. 

19.  Show  that  cos  a  •  see  a  does  not  depend  on  a.  Show  the  same  for 
csc2 «  -  eot^  a. 

20.  Does  sin2  a  +  eos^  a  depend  on  a  ?    Does  see^  a  —  tan^  a  ? 

21.  Before  reading  Art.  11,  find  the  values  of  the  sine,  cosine,  and  tangent 
of  30°,  45°,  and  60°. 

9.    Fundamental  relations.     The  Pythagorean  theorem 
a;2  -J-  ?/2  _  ^2^ 


FUNDAMENTAL   RELATIONS  9 

and  the  definitions  of  Art.  6  yield  certain  fundamental  relations 

between  the  six  trigonometric  functions  of  a  single  angle.  Thus, 
we  have  directly  from  the  definitions 

sma           .  ^  ^ 

sec  a  = -,  (2^ 

cos  a 

cota  =  - — -•  (3) 
tana 


Again,  by  division, 


tena  =  ^"^,  (4) 

cos  a 

and  cota=^^^.  (5) 

sin  a 

Dividing  by  v^  both  members  of  the  equation 

2/2  -f  ^2  _  ^2^ 

we  have  {t\  j^{^  ^x 

whence  sin^  a  +  cos^  a  =  1 .  (6) 

Dividing,  in  like  manner,  by  a^  and  by  y^  respectively,  we 
obtain 

tan^  a  +  1  =  sec^  a,  (T) 

cot^  a  +  1  =  csc^  a.  (8) . 

These  eight  equations  constitute  the  fundamental  relations  of 
trigonometry.  Of  these  the  fifth,  seventh,  and  eighth  may  be 
derived  algebraically  from  the  other  five.  By  means  of  these 
equations  it  is  possible  to  express  all  six  functions  in  terms  of  any 
one  of  them.  If  the  value  of  any  one  is  given,  the  values  of  the 
others  can  be  found.  Simple  numerical  examples  of  the  latter 
kind  are  more  quickly  solved  by  the  geometrical  method  of  Art.  7. 

Example  i.  To  find  the  remaining  functions  of  the  acute 
angle  whose  tangent  is  -f^. 

(1)  G-eometric  Method.  The  right  triangle  OMP  (Art.  6, 
^^^'  5)  has  sides  of  relative  length  ;«/  =  5,  a;  =  12,  whence  on  the 
same  scale  v  =  13.     Thus  the  sine  is  -j^,  etc. 


10  THE   ACUTE   ANGLE 

(2)  Analytic   Method.     Given  tana  =  ^5_^     Then  by  the  for- 
mulas of  Art.  9, 


1         12  ,. ^-      13. 

13 


cot  a  = =  —  ;    sec  a  =  Vl  +  tiin^  «  -  zj^ , 

tan «       5  12 


CSC  a  =  VI  +  cot^  CL—  r 


1        12 

cos  a  = =  —7 ;    sm  a 


sec  a      13  esc  a      18 


Example  2.    To  express  all  the  functions  of  a  in  terms  of 

sec «.     By  the  use   of  the   appropriate  formulas   of   Art.    9,  we 

obtain 

1  .  /^ 9  -      Vsec^  «  —  ]  ; 

cosa  = ;    sina  =  VI  —  cos^a  = 

sec  a  sec  a 

1  sec  a  

csca  =  -r— -  =  — =^=i'   tan«=  Vsec2«-l; 
sin  «      Vsec^a— 1 

cot  06  = 


tana      Vsec^a-l 

Example  3.    Verify  the  following  relation  by  reducing  the 
first  member  to  the  second  : 

tanSyg        1  o 

^    —  1  =  sec  /3. 


sec  /3  —  1 

By  means  of  the  formulas  of  Art.  9,  we  have 

tan'-^  ^        -<       sec2  yS  —  1^  o,ii  o 

W^  -1  = S — -- 1  =  sec/3+1 -1  =  sec ^. 

sec  p  —  1  sec  p  —  1 

EXERCISE  IV 

By  means  of  the  formulas  of  Art.   9,  find  the  values  of  the 
remaining  functions  of  each  of  the  following  angles,  given 

1.   sina  =  -iV5.         2.   cot/3  =  ff-         3.    cosy  =  ||.         4.   tany^^^. 

Express  all  the  functions  of  a  in  terms  of 

5.  tan  a.  6.   cos  a.  7    cot  a.  8.  sin  a. 

Find  the  values  of  the  following  expressions: 

ft    tan  a  +  cot  a       i  9 

9. ,  when  cos  a  =  — 

tana  —  cot  a  41 

,  r,    sec  a  —  cos  a       i         •  12 

10. ,  when  sm  a  =  — 

tan  a  —  sin  a  13 


FUNCTIONS  OF  COMPLEMENTARY  ANGLES 


11 


11.  ^    ^^"^^  +  cotB,  when  tanB=—- 
1  + cos/3  ^  ^      21 

12.  cos  ^  •  tan  /?  +  sin  y8  .  cot  y8,  when  sec  /?  =  ||. 
Express  the  following  in  terms  of  a  single  function 


13. 


CSC  a 


cot  a  +  tan  a 


in  terms  of  cos  a- 


,  -         sin  B       ,  1  +  COS  B  •     .  ^        ^ 

14. ^=-—  H =!^^ — —^  m  terms  of  esc  B. 

1  +  cos  /?  sni  y8  ^ 

15.  sec  y  —  tan  y  in  terms  of  sin  y. 
1  1 


16. 


+ 


1  —  sin  y      1  +  sin  y 

Verify  the  following  identities  : 

17.  cos4y8-sin4yS  =  2cos2^-l. 

18.  cos  a  '  tan  a  =  sin  a. 

cot2a 


in  terms  of  tan  y. 


19. 
20. 


cos'^  a. 


1  +  cot2  a 

^      + L_ 

tan^yg+l      cot2y8  +  l 


I. 


10.  Functions  of  complementary 
angles.  If,  in  Fig.  6,  Z  XOjS  is 
constructed  equal  to  Z.AOY^ 
XOB  (=fi}  and  XOA  (=  a)  are 
complementary.  The  triangles 
OM'F'  and  Oi)[fP  are  similar,  the 
pairs  of  corresponding  sides  being 
v'  and  v^  x'  and  «/,  ?/'  and  x. 

We  have  then 

sin  (90°- a)  =  sin  ^  = 


cos  «, 


cos  (90°  —  a)  —  cos  ^  =    ,  =  ^  =  sin  a, 

v'       V 


tan  (90°  -  «)  =  tan  ^  =  -^^  =  -  =  cot  «, 

cot  (90°-  a)  =  cot  /?=  -.  =  ^=  tan  a, 
y      ^ 


12 


THE   ACUTE   ANGLE 


sec  (90°  -  «)  =  sec  /?  =  -^ 


CSC  a, 


CSC  (90' 


a)  =  CSC  8=  —.  =  -=  sec  a. 


The  prefix  "  co "  is  thus  seen  to  be  the  abbreviation  of  the 
word  "complement's."  The  general  theorem  may  be  stated  as 
follows : 

Ant/  trigonometric  function  of  an  acute  angle  is  equal  to  the 
corresponding  co  function  of  its  complementary  angle. 

Thus,  sin  72°  =  cos  18°,  cot  54°  =  tan  36°,  etc. 

11.  Functions  of  45°,  30°,  60°.  The  exact  values  of  the  func- 
tions of  certain  angles  are  readily  found. 

(1)  If,  in  Fig.  7,  Z  XOA  =  45°,  the  triangle  OMF  is  isosceles, 


so  that  x  —  y  = 


We  have  at  once 


>X 


V2 

sin  45°  =  cos  45°  =  \-\/%. 
tan  45°  =  cot  45°  =  1, 
sec  45°  =  CSC  45°  =  V2. 


>X 


Fig.  7. 

(2)  If,  'in  Fig.   8,  AXOA  =  ^0\   and       "{Y 
/.  XOQ    is    constructed    equal    to    —30°, 
the  triangle   QOP  is   equilateral,   so  that 
y  =  ^v,  x  =  \^lv. 

We  have,  at  once, 

sin  30°  =  |,  cos  30°  =  I V3, 
tan  30°  =  1 V3,  cot  30°  =  V3,. 
sec  30°=|V3,  CSC  30°  =  2. 

(3)  In  like  manner  to  (2),  or  by  Art.  10,  we  obtain 

sin  60°=  J  V3,  COS  60°  =1, 

tan  60°  =  V3,  cot  60°  =  l  VB, 

sec  60°  =  2,  CSC  60°  =  | V3. 
12.    Functions  of  0°  and  90°. 

(1)  As  the  Z  XOA  (see  Fig.  9)  decreases  so  as  to  approach 
the  limit  zero,  the  abscissa  x  will  approach  equality  to  the  radius 


FUNCTIOA^S   OF   0°  AND   QO'^ 


13 


vector  V.     If,  at  the  same  time,  the  radius  vector  remains  finite, 
the  ordinate  ?/  will  approach  the  limit  zero. 

It  should  be  noticed  that  the  cosecant  varies  in  such  a  manner 
that  its  denominator  approaches  the  limit  zero  while  its  numera- 
tor remains  constantly  equal  to  the 
finite  number  v,  so  that  the  value 
of  the  cosecant  increases  without 
limit  as  the  angle  approaches  zero. 
It  is  then  said  to  become  infinite 
and  is  represented  by  the  symbol  oo. 
The  cotangent  also  becomes  infinite 
as  the  angle  approaches  zero,  since 
its  numerator,  although  changing, 
never  exceeds  v. 

We  have,  then,  sin  0°  =  0, 

tan  0°  =  0, 
sec  0°  =  1, 


(2)  In  like  manner,  we  obtain 

sin  90°  =  1,     cos  90°  =0, 
tan  90°=  00,    cot  90°  =  0, 

sec  90°  =  00,   CSC  90°  =  1. 

Example.      Solve  the  equation 
sec^  7  —  V3  tan  7  =  1. 

Expressing  wholly  in  terms  of  tan  7, 
tan^  ry  -f  1  —  V3  tan  7—1  =  0, 
tan^  7  —  V3  tan  7  =  0, 
tan  7=0  and  V3. 


■^1 

1^ 

p 

(2) 

'1 

1 

y 

• 

I 

0 

J 

\i 

X 

Fig.  10. 


we  have 

or 

whence 


Then,  by  Arts.  11  and  12, 

7  =  0°  and  60°. 


EXERCISE  V 

1.  Express  in  terms  of  an  angle  less  than  45°  the  functions  of  75". 

2.  Express  in  terms  of  an  angle  less  than  45°  the  functions  of  65°. 


14  THE   ACUTE   ANGLE 

Verify  the  following  for  «  =  30° ;  also  for  a  =  45° ; 

3.  sin  2  a  =  2  sin  a  cos  a. 

4.  cos  2  a  =  cos^  a  —  sin^  a. 
2  tan  a 


5.  tan2a  = 

6.  cot  2  a  = 


1  —  tan^  a 
cot2  a-\ 


2  cot  a 
Notice  that  sin  2  a  does  not  equal  2  sin  a,  cos  3  a  does  not  equal  3  cos  a,  etc. 

Verify  for  a  =  30°  : 

7.  sin  3  a  =  3  sin  a  —  4  sin*  a. 

8.  cos  3  a  =  4  cos^  a  —  3  cos  a. 

Evaluate  the  following  expressions  : 

9.  sin  60°  cos  30°  -  cos  60°  sin  30°. 

10.  cos  60°  cos  30°  +  sin  60°  sin  30°. 

11.  csc2  45°  +  sin  60°  tan  30°. 

12.  sin  60°  tan  45°  -  sec  30°  sin2  45°  cot  60°. 

Solve  each  of  the  following  equations  for  some  one  function  of 
a  and  find  the  angle  in  degrees.  Verify  the  results  by  substitu- 
tion in  the  given  equation. 

13.  tan  a  -  3  cot  a  =  0. 

14.  sec  a  +  2  cos  a  =  3. 

15.  4  sin2  a  +  3  cot^  a  =  4. 

16.  since  +  3cosa  =  2V2. 

17.  A  ladder  22  feet  long  leans  against  a  wall,  its  foot  being  11  feet  away 
from  the  wall.  Find  (a)  the  angle  formed  by  the  ladder  with  the  ground; 
(6)  the  height  of  the  top  of  the  ladder  above  the  ground. 

18.  The  diagonal  of  a  rectangle  makes  an  angle  of  30°  with  the  long  side. 
If  the  length  of  this  side  is  14  inches,  what  is  the  length  of  the  short  side  of 
the  rectangle  ?  of  the  diagonal  ? 

19.  The  side  of  a  conical  pile  of  sand  makes  an  angle  of  45°  with  the 
floor.     If  the  height  from  the  floor  is  12  feet,  what  is  the  area  of  the  base  ? 

20.  A  guy  rope  (assumed  to  be  straight)  has  a  length  of  60  feet  and 
extends  from  the  top  of  a  mast  30  feet  high  to  the  ground.  Find  the  angle 
between  the  rope  and  the  mast. 

13.  Variation  of  the  trigonometric  functions  as  the  angle  varies. 
Suppose  the  angle  6  to  vary  continuously  from  0°  to  90°.     The 


VARIATION.    INVERSE  FUNCTIONS  15 

terminal  side  revolves  about  the  origin  from  the  position  OX  to 

the  position  OY.     li  v  remains  constant,  y  will  increase  from  0 

to  V,  while  X  will  decrease  from  v  to  0.     Consequently,  the  nu- 

y 
merator  of  the  fraction -(=  sin  ^)  increases  from  0  to  v.  while  the 

V 

denominator  remains  constant.  Hence,  while  6  increases  from 
0°  to  90°,  sin  6  increases  from  0  to  1. 

The  numerator  of  the  fraction  -  (  =  cos  6)  decreases  from  v  to 

V 

0,  while  the  denominator  remains  constantly  equal  to  v.  Hence, 
while  0  increases  from  0°  to  90°,  cos  0  decreases  from  1  to  0. 

The  numerator  of  the  fraction  -(=  tan  0^  increases  from  0  to 

X  ^ 

V,  while  the  denominator  decreases 
from  V  to  0.  Hence,  while  6  in- 
creases from  0°  to  90°,  tan  ^,  be- 
ginning with  zero,  increases  with- 
out limit  as  6  approaches  90°. 
We  express  this  by  saying  that 
tan  6  varies  from  0  to  qo. 

The  student  should  trace  care- 
fully the  variation  of  the  other 
trigonometric  functions  and  compare  the  results  with  the  values 
found  in  Arts.  11  and  12.  Article  7  should  be  read  again  at  this 
point. 

14.  Inverse  trigonometric  functions.  The  same  functional  re- 
lation is  expressed  by  the  two  statements,  "  m  is  the  sine  of  the 
acute  angle  a"  and  "a  is  the  acute  angle  whose  sine  is  m."  The 
corresponding  symbolic  notations  are 

m  =  sin  a,  a  =  arcsin  m,* 

with  the  understanding  that  a  is  an  acute  angle  and  that  m  is  a 
positive  number  not  greater  than  unity.  A  similar  symbolic 
relation  holds  for  the  other  trigonometric  functions.  It  is  fre- 
quently read  "  arc-sine  m,"  or  "anti-sine  m,"  since  two  mutually 
inverse  functions  are  said  each  to  be  the  anti-function  of  the  other. 

*  This  notation  is  universally  used  in  Europe  and  is  fast  gaining  ground  in  this 
country.     A  less  desirable  symbol, 

a  =  sin-i  m, 
is  still  found  in  English  and  American  texts. 

The  notation  a  =  inv  sin  m  is  perhaps  better  still  on  account  of  its  general 
applicability.     (See  Art.  80.) 


16  THE   ACUTE   ANGLE 

The  inverse  notation  is  convenient  for  the  statement  of  prob- 
lems.    The  purposes  of  interpretation  and  manipulation  are  better 
served  by  transforming  to  the  corresponding  direct  notation. 
Example.     Find  the  value  of  sin  (90°  —  arccot  ^^2)- 
In  the  direct  notation  the  example  reads :   Given  cot  a  =  -^^^ 
find  sin  (90°  -  a).     Then,  by  Arts.  10  and  9, 

sin  (90°  —  a)  =  cos  a  =  ^\. 

EXERCISE  VI 

1.  Trace  the  variation  in  value  of  sec  6. 

2.  Trace  the  variation  in  value  of  esc  0. 

Find  the  values  of  the  following : 

3.  tan  (cos-iyV)-  7.  sec  (90°- arcsec  2). 

4.  sin  (arccot  |).  8.  esc  (90°—  arccsc  V2). 

5.  cos  (90°-arctan  ^\).  9.  sin  (2  tan-i  1). 

6.  cot  (90°-  sin-i  if).  10.  cos  (2  sin-i  I). 

Solve  the  following  equations : 

11.  2sin2/?  +  3cos/8-3  =  0. 

12.  sec  y8  -  2  tan  )8  =  0. 

13.  tan^(2sin^-l)(secj8-V2)  =  0. 

14.  sin  y8  (2  cos  j8  -  V'3)  (tan  yS  -  1)  =  0. 

Verify  the  identities : 

15.  sin^  a  +  cos*  a  +  sin^  a  cos^  a  =  sin*  a  —  sin^  a  +  1. 

16.  (esc  a  —  cot  a)  (esc  a  +  cot  a)  =  1. 

17.  (tan  «  + cot  a)  (sin  a -cos  a)  =  1. 

18.  1  -  tan*  a  =  2  sec^  a  -  sec%. 

19.  sin«  a  +  cos^  a  =  1  -  3  sin^  a  cos^  a. 

20.  cos* a  -  sin* a  =  l  -2  sin^ a. 

15.  Orthogonal  projection.  In  accordance  with  the  definitions 
of  Art.  4  (see  note,  page  4)  it  follows  that  the  projection  of  a 
line  segment  on  any  line  is  equal  to  the  length  of  the  line  segment 
multiplied  by  the  cosine  of  the  angle  formed  by  the  line  segment 
with  the  line  of  projection.  Thus,  in  Fig.  12,  the  projection  of 
AB  on  RS  is 

M]V=  AB  cos  a. 


ORTHOGONAL  PROJECTION 


IT 


In  like  manner,  the  projection  of  AB  on  a  line  perpendicular 
to  MS  (i.e.  making  +  90°  with  MS)  has  the  value  AB  sin  a. 
These  projections  are  called  the  components  of  the 
line  segment  AB  along  and  at  right  angles 
to  the   direction   MS. 

In  physics,  line  segments  are 
often  used  to  represent  quanti- 
ties that  have  direction  as  well  as 
magnitude ;  for  example,  forces, 
velocities,  accelerations.  The  components  of  the  line  segment 
used  to  represent  a  force  represent  components  of  the  force ;  like- 
wise for  a  line  segment  representing  a  velocity,  acceleration,  or 
moment.  Suppose,  for  example,  that  the  line  segment  AB,  Fig. 
13,  represents  a  force  applied  to  the  block  m  resting  on  a  liorizon- 
tal  plane.     This  segment  has  the  component  UB  parallel  to  the 

plane  and  the  compo- 
nent FB  perpendicular 
to  the  plane.  Segment 
FB  represents  a  force 
component  F^  parallel 
to  the  plane,  which 
tends  to  move  the  block 
along  the  plane  ;  seg- 
ment FB  represents  a 
force  component  F.^  per- 
pendicular to  the  plane  and  tending  to  produce  pressure  between 
the  block  and  plane.  Denoting  by  F  the  force  represented  by 
AB,  we  have 


F^  =  F  cos  a, 


Fy  =  F  sin  a. 


Example.  At  a  given  instant  a  point  is  moving  in  a  direc- 
tion at  an  angle  of  30°  with  a  given  horizontal  line  with  a  velocity 
of  20  feet  per  second.  Find  the  component  of  the  velocity  along 
the  line. 

Taking  the  given  line  as  the  JT-axis,  we  have  for  the  compo- 
nent v^ 

v^  =  v  COS  30°=  20  X  I  V3  =  17.321  feet  per  second. 

The  component  along  a  line  perpendicular  to  the  given  hori- 
zontal line  in  the  plane  of  motion  is 


Vy  =  v  sin  30°=  20  x  |  =  10  feet  per  second. 


18  THE   ACUTE   ANGLE 

EXERCISE  VII 
The  student  should  draw  appropriate  figures  for  each  of  the  following  exercises. 

1.  Find  the  projections  of  a  line  segment  8.5  inches  in  length  on  the  X- 
and  Z-axes,  (a)  when  the  segment  makes  an  angle  of  45°  with  the  Z-axis ; 
(b)  when  it  makes  an  angle  of  60°  with  the  F-axis. 

2.  A  crank  16  inches  long  rotates  in  a  vertical  plane.  When  the  crank 
makes  an  angle  of  30°  with  the  horizontal  diameter  of  the  circle  described  by 
the  moving  end,  what  is  the  distance  of  the  moving  end,  (a)  from  the  hori- 
zontal diameter?  (h)  from  the  vertical  diameter  ? 

3.  If  in  Fig.  13  the  force  jP  denoted  by  AB  is  40  pounds,  find  the  compo- 
nents F^  and  Fy,  (a)  when  a  =  30° ;  (b)  when  a  =  45°.  Discuss  the  cases 
a  =r  0°  and  a  =  90°. 

4.  A  steamer  is  moving  at  a  speed  of  18  miles  per  hour  in  a  direction 
north  of  east,  making  an  angle  of  30°  with  an  east  and  west  line.  At  what 
rate  is  the  steamer  sailing  eastward  ?  at  what  rate  northward  ? 

5.  A  guy  wire  exerts  a  pull  of  3000  pounds  on  its  anchorage  and  makes  an 
angle  of  30°  with  the  ground.  Find  the  component  of  this  force,  (a)  along 
the  ground;   (b)  vertical. 

6.  The  eastward  and  northward  components  of  the  velocity  of  a  moving 
body  are  found  to  be  1^^=  12  miles  per  hour  and  Vj,=  12  V3  miles  per  hour,  respec- 
tively.    Find  (a)  the  magnitude  and  (b)  the  direction  of  the  body's  velocity. 


CHAPTER   ITT 


RIGHT  TRIANGLES 


16.  Laws  for  solution.  If,  in  a  right  triangle,  two  independent 
parts  are  known,  in  addition  to  the  right  angle,  the  three  remain- 
ing parts  can  be  found.  Thus  two  given  parts,  at  least  one  of 
which  is  a  side,  determine  a  right  triangle.  The  formulas  needed 
in  all  cases  to  effect  this  solution  are 
five  in  number.  Two  are  the  state- 
ments of  well-known  geometric  theo- 
rems, while  the  other  three  are  the 
immediate  consequences  of  the  defini- 
tions of  the  trigonometric  ratios  con- 
tained in  Art.  6. 

In  Fig.  14  let  ACB  be  a  right 
triangle,  right-angled  at  0.  We  shall 
denote  the  interior  angles  at  the  ver- 
tices by  a,  yS,  7,  and  the  lengths  of 
the  sides  opposite  them  by  a,  5,  c, 
respectively.      Note    that   <y  =  90°,    and 

The  five  formulas  are  the  following : 


B 

/f^ 

X 

a 

X 

y 

A 

b 

c 

-' 

C     IS 


Fig.  14. 

the   hypotenuse. 


(1) 

(2) 
(8) 

(-1) 

(5) 

Equation  (1)  follows  from  the  Pythagorean  theorem,  and 
(2)  from  the  fact  that  the  sum  of  the  angles  of  a  triangle  is  equal 
to  two  right  angles.  In  order  to  establish  the  last  three,  place 
the  triangle  on  the  axes  of  coordinates  described  in  Art.  4,  the 
side  JL(7  extending  from  the  origin  to  the  right  along  tlie  X-axis, 
and  the  hypotenuse  lying  in  the  first  quadrant,  as  in   Fig.  14. 

19 


a2+62_ 

c\ 

a+p  = 

90*^ 

» 

a 

c 

=  sin  a  = 

cos 

p. 

h 
c 

=  cos  a  = 

sin 

p. 

a 
h 

=  tan  a  = 

cot 

p- 

20  RIGHT   TRIANGLES 

Then  5,  a,  c,  are  respectively  the  abscissa,  ordinate,  and  radius 
vector  of  j5,  a  point  on  the  terminal  side  of  the  angle  a,  which  is 
conventionally  placed. 

It  follows  at  once  from  Art.  6  that 

a 

-  —  sm  a, 
c 

h 

-  =  cos  a,  • 

a      , 

-  =  tan  a. 

0 

The  corresponding  values  of  the  functions  of  the  angle  y8  result 
from  Art.  10. 

17.  Area  of  right  triangles.  The  formulas  for  'the  area  of  a 
right  triangle  follow  from  tiie  familiar  geometric  theorem 

Area  =  |^  x  base  x  altitude, 

or  expressed  symbolically, 

A  =  \ab.  (1) 

The  substitution  for  a  of  its  value  from  the  preceding  article  gives 

^  =  1  6c  sin  a.  (2) 

Again,  introducing  the  values  of  both  a  and  5, 

A  =  I  c2  sin  a  •  cos  a.  (2) 

Other  formulas  for  the  area  may  also  be  obtained. 

18.  Method  of  solution.  The  solution  of  any  problem  consists 
of  four  parts :  the  analysis,  the  algebraic  solution,  the  arithmetical 
computation,  and  the  interpretation  of  the  results. 

(1)  The  student  should  read  and  analyze  the  problem,  noting 
which  parts  are  known  and  which  are  desired.  The  construction 
of  a  neat  and  sufficiently  accurate  figure  is  helpful  and  advisable. 

(2)  The  student  should  select  from  the  five  formulas  of  Art. 
16  those  containing  a  single  unknown  part  each,  in  addition  to  the 
known  parts,  and  should  solve  them  for  these  unknown  parts  while 
still  in  the  literal  form. 

Experience  has  led  to  the  adoption  of  the  following  two  rules 
of  procedure : 


SOLUTION   OF   EIGHT   TRIANGLES  21 

(tI)  The  use  of  the  Pythagorean  formula,  a^  -f  5^  =  c^,  is  to  be 
avoided  save  when  the  data  are  very  simple  or  a  table  of  squares 
and  square  roots  is  at  hand. 

(^)  So  far  as  is  consistent  with  rule  A,  each  unknown  part 
should  be  found  in  terms  of  those  parts  originally  given  in  the 
problem,  in  order  to  avoid  accumulation  of  errors. 

In  conformity  with  these  rules,  the  angle  relation  a-\- 13  =  90°, 
and  two  of  the  three  trigonometric  formulas  serve  to  effect  the 
solution,  while  the  remaining  trigonometric  formula  affords  a 
check  on  the  work. 

(3)  The  solution  is  now  effected  by  introducing  the  numerical 
data  and  performing  the  necessary  computations.  The  correctness 
and  accuracy  of  the  results  are  greatly  enhanced  by  extreme  order- 
liness of  arrangement  and  neatness  of  detail. 

The  use  of  the  trigonometric  tables  and  the  employment  of 
suitable  checks  will  be  discussed  in  subsequent  articles. 

(4)  The  geometric  or  physical  significance  of  the  results  ob- 
tained should  be  fully  considered  and  interpreted. 

Example  i.    Given  c  ==  254,  a  =  30°,  to  find  a,  5,  /3. 

In  this  instance  the  analysis  and  construction  are  obvious. 

The  three  appropriate  formulas  yield  at  once  the  forms 

yS=90°-a, 

a=  c  sin  ct, 

h  =  c  cos  a. 

The  formula  b  =  a  tan  /3  affords  the  check. 

On  introducing  the  numerical  data,  we  obtain 

^  =  90°  -  30°   =  60°, 

a  =  25ixi       =254  X. 5  =  127, 

^>  =  254  X  1V3  =  254  X  .86605  =  219.976T. 

The  check  formula  gives 

6  =  127x1.7321  =  219.9767. 

Example  2.  Given  «  =  39.00,  6  =  80.00,  to  find  c,  a,  /9,  and 
the  area. 

As  before,  we  may  pass  immediately  to  the  second  stage.  Now 
c  is  given  directly  in  terms  of  a  and  b  by  the  formula  c?=  Va^  +  h^ 


22  RIGHT   TRIANGLES 

If  we  are  to  avoid  the  use  of  this  formula,  we  must  first  find  a  and 
/8,  and  then  get  c  by  means  of  one  of  these  angles.  We  use  the 
forms : 


I  a  = 

a 

"V 

/9  = 

90°- 

C  = 

a 

sm  a 

A  = 

\ah. 

c  = 

h 
sin  l3' 

and  for  the  check 

We  obtain,  then  tan  «  =  39  -^  80  ==  .4875, 

a  =  25°  59^  as  found  from  Table  III, 
/3=  90° -25°  59' =64°  01', 
^  =  39  _j.  .4381  =  89.01, 
^  =  1  X  39  X  80  =  1560, 

and  for  the  check  c=SO-^  .8989  =  89.00, 

showing  a  difference  of  .01. 

On  account  of  the  simplicity  of  the  numbers,  we  may,  by  using 
the  formula  c^  =  a^-]-  P,  find,  exactly,  c  =  89. 

Explain  the  accumulation  of  errors  and,  hence,  the  reason  for 
rule  of  procedure  (jB). 

Examples,  l.    Given  c  =  42,  ^  =  arcsin  .28  ;  find  a  and  h, 

2.  Given  5  =  27,  «  =  tan~i  .75 ;  find  a  and  c. 

3.  Given  a  =  300,  a  =  cos"^  .45  ;  find  c  and  b. 

4.  Given  c=  200,  a—  arccot  1.12;  find  a  and  h. 

19.  Trigonometric  tables.  In  the  first  example  worked  in  the 
preceding  article,  the  functions  of  30°  had  been  determined  in 
Art.  11.  In  the  second  example,  however,  the  value  of  tan  a  was 
not  one  of  those  previously  ascertained,  and  the  value  of  a  was  not 
recognizable  from  its  tangent.  For  convenience  of  reference  the 
numerical  values  of  the  sines,  cosines,  tangents,  and  cotangents  of 
all  angles  differing  by  intervals  of  one  minute  from  0°  to  90°  have 
been  collected  in  Table  III,  on  pages  71-89.  The  arrangement  is 
simple  and  plain.  The  degree  numbers  from  0°  to  44°  occur  at  the 
top  of  the  page,  with  the  minutes  running  down  the  left  margin. 


TABLES.  ERRORS  AND  CHECKS  23 

The  numerical  values  of  the  functions,  computed  to  four  decimal 
places,  are  placed  in  columns  under  the  names  of  the  functions. 

Since  sin  (90°  —  a)—  cos  a,  and  tan  (90°  —  ce)  =  cot  a,  the  space 
required  may  be  reduced  one  half.  The  degree  numbers  of  angles 
from  45°  to  90°  are  printed  at  the  bottom  of  the  pages  in  reversed 
order,  the  minutes  run  up  the  right  margin,  and  the  names  of  the 
functions  are  in  reversed  order  at  the  bottom. 

For  the  present  the  student  need  not  concern  himself  with 
smaller  divisions  of  the  angle  than  the  minute.  Further  refine- 
ment is  attained  by  a  method  to  be  described  in  Art.  26. 

Table  IV,  on  pages  91-93,  contains  the  squares  of  numbers  less 
than  1000  and,  by  interpolation,  of  numbers  up  to  9999.  The 
first  page  gives  directly  the  squares  of  numbers  from  1  to  100. 
On  the  second  and  third  pages  the  tens  and  units  digits  of  the 
number  to  be  squared  are  in  the  left  margin,  while  the  hundreds 
digits  are  at  the  tops  of  the  several  columns.  The  last  two  figures 
of  the  square  are  in  the  column  at  the  right  under  U.,  opposite 
the  tens  and  units  digits ;  the  first  three,  or  four,  figures  of  the 
square  are  in  the  same  line  in  the  column  under  the  hundreds 
digit.  In  the  right  margin  are  the  last  two  figures  of  the  tabular 
difference  used  in  interpolation,  to  which  must  be  prefixed  the 
remainder  obtained  by  subtracting  the  first  three,  or  four,  figures 
of  the  square  from  those  in  the  same  column  immediately  beneath, 
or  that  remainder  diminished  by  1  when  the  asterisk  (*)  is  present. 
The  use  of  the  table  is  best  shown  by  illustration. 

Examples,     i.    3282  =  107,584. 

2.  475.3  =  4752  4-  .3  X  951  =  225,625  +  285  =  225,910.* 

3.  28.372  =28.32  +  . 07  x567  =  800.89 +  3.97  =  80 1.56. 

Square  roots  are  extracted  by  reversing  the  process ;  thus. 


4.    V27556  =  166. 


5.    V658,037  =  V657,72l  +  316  ^  1623  =  811  +  .2  =  811.2. 

20.  Errors  and  checks.  The  results  obtained  are  not  always, 
nor  even  usually,  exactly  correct.  The  deviations  from  the  true 
values  are  of  two  sorts,  mistakes  and  errors,  and  a  sharp  distinc- 
tion must  be  made  between  them. 

*  This  result  is,  of  course,  only  approximately  correct.  The  true  result  may  be 
obtained  as  follows : 

475.32  =  4752  +  .3  X  (475.3  +  475)  =  225,625  +  285.09  =  225,910.09. 


24  RIGHT   TRIANGLES 

The  data  for  problems  arising  in  actual  practice  are  derived 
from  observations  made  with  instruments  for  measuring  lengths, 
angles,  etc. 

Mistakes  may  arise  from  a  false  reading  of  the  observing  instru- 
ment, a  misapprehension  of  the  problem,  the  employment  of  the 
wrong  formula,  faulty  addition,  etc.  ^  They  are  never  allowable  or 
excusable. 

On  the  other  hand,  instruments  are  so  constructed  as  to  yield 
results  only  to  a  certain  degree  of  precision,  which  should  be 
ascertained  for  each  instrument.  Moreover,  observation  is  per- 
formed by  the  human  apparatus,  eyes,  ears,  etc.,  and  a  certain  per- 
sonal equation,  an  anticipation  or  lagging  in  sight  or  hearing,  is 
always  present,  varying  with  personal  fitness  and  experience. 
Methods  of  eliminating  instrumental  errors,  so  as  to  obtain  the 
maximum  precision  possible  with  the  instruments  used,  are  given 
in  standard  works  on  engineering  instruments.  Again,  the  arith- 
metical calculation  involves  the  trigonometric  ratios,  which  are,  in 
general,  non-terminating  decimal  fractions,  while  their  values  in 
the  mathematical  tables  are  computed  only  to  a  certain  number  of 
decimal  places.  Errors,  therefore,  will  always  be  present ;  but 
every  precaution  should  be  taken  to  keep  the  errors  due  to  com- 
putation well  within  the  limits  of  error  of  the  observed  data  and 
desired  results  fixed  by  the  nature  of  the  problem. 

In  both  observation  and  solution,  certain  additional  processes 
are  employed,  to  avoid,  or  to  reveal,  mistakes.  These  processes 
are  known  as  checks  and  vary  with  the  nature  of  the  problem. 

While  no  general  rules  for  checks  can  be  laid  down,  a  frequent 
practice  in  the  solution  of  triangles  is  to  make  use  of  a  formula 
connecting  the  required  parts,  just  found,  noting  if  the  results  are 
within  the  range  of  allowable  error.  The  size  of  this  allowable 
error  should  be  known  for  each  table. 

As  a  check  to  arithmetical  computation,  graphical  construction 
is  well  understood  and  strongly  advised.  As  a  means  of  avoiding 
the  grosser  mistakes,  a  free-hand  sketch  will  frequently  suffice  by 
guiding  the  student  to  a  reasonable  interpretation  of  data,  and 
indicating  possible  results. 

A  drawing  constructed  to  scale  will  further  aid  by  yielding 
values  more  or  less  approximate,  approaching  those  obtained  by 
computation. 

Carried  a  step  farther  as  regards  accuracy,  by  the  use  of  pre- 
cise instruments,  the  graphical  construction  often  attains  to  the 


PROBLEMS  INVOLVING   RIGHT   TRIANGLES 


25 


dignity  of  an  independent  solution,  with  results  falling  within  the 
limits  prescribed  by  the  physical  conditions  of  the  problem. 

There  is  no  better  evidence  of  careful  work  than  the  record- 
ing of  a  reasonable  error  obtained  by  the  comparison  of  two 
methods.  In  practical  work  the  allowable  per  cent  of  error 
becomes  an  important  consideration. 


EXERCISE  VIII 


Find  the  missing  parts  of  the  following  triangles,  using  the 
natural  trigonometric  functions,  Table  IIL 


a 

/3 

a 

h 

c 

A 

1. 

2.5°  10' 

34 

2. 

52°  20' 

73 

3. 

61°  15' 

243 

4. 

78°  35' 

521 

5. 

21°  25' 

235 

6. 

72°  45' 

720 

7. 

80°  30' 

1200 

8. 

17°  30' 

1500 

9. 

240 

360 

10. 

381 

715 

11. 

521 

630 

12. 

840 

1400 

13. 

648 

864 

14. 

595 

600 

15. 

215 

385 

16. 

2111 

1234 

17. 

95 

7980 

18. 

264 

30360 

19. 

74°  20' 

1225 

20. 

24°  50' 

843 

21.   In  the  same  vertical  plane  the  distances  shown  in  Fig.  15  were  meas- 
ured in  feet  along  the   surface  of 

the  ground.     The  distances  of  the     «    ^  ^         ^         ^ f^R- 

different  points  below  the  instru- 
ment, as  measured  by  a  rod,  are 
given  also  in  feet.  The  vertical 
scale  is  exaggerated  for  clearness. 
What  is  the  horizontal  distance 
from  B  to  Gl     (Check  by  a  table    of   squares    and   square   roots.) 


26  RIGHT   TRIANGLES 

22.  A  line  surveyed  across  a  ridge  is  1500  feet  in  horizontal  length. 
Stakes  are  set  100  feet  apart  horizontally  by  level  chaining.  By  leveling,  the 
elevations  of  the  surface  at  the  different  stakes  is  obtained  as  follows :  730.2, 
735.9,  739.7,  743.4,  750.1,  751.8,  760.7,  764.1,  764.3,  765.8,  765.0,  763.2,  758.3, 
750.2,  743.1,  740.2.  What  length  of  wire  will  be  required  for  fencing  along 
this  line?     (Check  by  a  table  of  squares  and  square  roots.) 

23.  If  a  gravel  roof  slopes  one  half  inch  to  the  horizontal  foot,  what  angle 
does  it  make  with  the  horizon? 

24.  If  the  face  of  a  wall  has  a  batter  or  inclination  of  one  inch  in  one  ver- 
tical foot,  what  is  its  angle  with  the  vertical  ? 

25.  What  is  the  angle  of  ascent  of  a  railway  built  on  a  2  per  cent  grade 
(i.e.  2  vertical  feet  to  100  horizontal  feet)  ? 

C  26.    The  pitch  of  a  roof  is  the  ratio 

— .      (See   Fig.   16.)     What  is  the   in- 

clination   to  the  horizon  of   a  roof   with 
•^  pitch,  I  pitch,  I   pitch? 

27.   What  is  the  pitch  of  a  roof  slop- 
ing to  the  horizon  at  15°,  30°,  45°  ? 

28.  What  is  the  inclination  to  the  horizon  of  the  corner  or  hip  rafter  of  a 
pyramidal  roof  whose  pitches  are  ^? 

29.  What  is  the  inclination  from  the  vertical  of  the  corner  edge  of  a  wall, 
both  of  its  faces  having  a  batter  of  ^^^  ? 

30.  At  what  angle  does  a  railway  slope  if  it  has  a  grade  of  0.25%,  0.5%, 

2.5%? 

31.  At  what  angle  must  a  cog  railway  ascend  in  order  to  rise  2640  feet  in 
one  horizontal  mile  ? 

32.  A  battleship  known  to  be  341  feet  long  subtends  an  angle  of  3°  20' 
when  presenting  its  broadside  to  a  fort  on  shore.  For  what  distance  should 
guns  be  sighted  when  trained  upon  it  ?  (Note  that  the  isosceles  triangle  hav- 
ing the  length  of  the  ship  for  its  base  is  separable  into  two  right  triangles.) 

33.  In  planning  the  stairway  for  a  house  it  is  desired  that  the  riser,  or 
vertical  distance  between  steps,  shall  be  7  inches,  and  the  treads,  or  horizontal 
distances  between  faces,  11  inches.  What  will  be  the  angle  of  inclination  of 
the  hand  rail? 

34.  Taking  the  data  of  the  preceding  problem,  what  will  be  the  length  of 
the  hand  rail  if  straight,  provided  the  height  between  floors  is  11  feet  8  inches? 

35.  A  cylindrical  water  tower  whose  external  diameter  is  25  feet  subtends 
a  horizontal  angle  of  5°  30'  as  viewed  from  a  distance.  How  far  is  its  center 
from  the  instrument? 

(Note  that  we  have  a  triangle  that  is  right-angled  when  the  line  of  sight 
is  tangent.  The  base  is  the  radius  of  the  tower  and  the  opposite  angle  is  half 
of  the  one  observed.) 


PROBLEMS   INVOLVING   RIGHT   TRIANGLES 


27 


36.  What  horizontal  angle  would  be  subtended,  at  a  distance  of  2  miles^ 
by  a  vertical  cylindrical  gas  receiver  60  feet  in  diameter  ? 

(See  note  to  problem  35.) 

37.  The  end  of  a  pendulum  34 inches  long  swings  through  an  arc  of  3|  inches. 
Find  the  angle  through  which  the  pendulum  swings. 

38.  When  vertically  over  a  village,  a  balloon's  angle  of  inclination,  as 
viewed  from  9  miles  distant,  was  15°  20'.  Assuming  the  surface  of  the  country 
to  be  fairly  level,  what  was  the  height  of  the  balloon  ? 

39.  A  flagstaff  110  feet  high  is  covered  by  a  vertical  angle  of  12°  30'  at  a 
point  approximately  on  a  level  with  its  center.  How  far  is  the  observer  from 
the  staff? 

40.  The  data  of  a  preliminary  survey  are  as  follows: 


AB  =  240.9  feet. 
BC  =  310.7  feet. 
CZ>  =  611.5  feet. 
DE  =  237.2  feet. 
J5:i^=  528.0  feet. 

Considering  A,  Fig.  17,  as  the 
origin  of  coordinates  and  AB  a,s 
the  axis  of  abscissas,  it  is  required 
to  compute  coordinates  for  all 
points  given,  thus  providing  for 
the  accurate  mapping  of  the 
survey. 

41.  Find  the  missing  parts 
and  area  of  the  following  isos- 
celes triangles  (see  Fig.  18  for 
lettering) 


Angle  at  5  =  62°  11'  left. 
Angle  at  C  =  55°  50'  left. 
Angle  at  D  =  43°  42'  right. 
Angle  a.tE  =  51°  23'  right. 


Fig.  17. 

35°,         a  =  42; 

«  =  72°,  &  =  12o; 

350,        &  =  180; 

/3  =  54°,  a  =  360 ; 

51°  26',  &  =  480; 

a  =  640,  b  =  840. 

42.  Find  the  lengths  of  the  chords  of  the  follow- 
ing arcs  in  terms  of  the  radius:  30°,  36°,  40°,  45°,  60°, 
75°,  90°,  120°.     Compute,  given  R  =  100. 

43.  Express  in  terms  of  the  sine  and  radius  the  relation  between  the  chord 
of  an  arc  and  the  chord  of  half  the  arc. 

44.  Express  in  trigonometric  form  the  most  important  relations  between 
the  radius  R  of  the  circumscribed  circle,  the  radius  r  of  the  inscribed  circle,  the 
side  s,  and  the  number  of  sides  n  of  a  regular  polygon. 


28  RIGHT   TRIANGLES 

45.  Compute  and  tabulate  the  perimeter  and  the  circumferences  of  the 
circum-  and  in-circles  of  a  regular  polygon  of  n  sides  for  n  =  4,  8,  16,  32,  given 
72  =  10. 

46.  Compute  and  tabulate  the  area  of  a  regular  polygon  of  n  sides  and  of 
its  circum-  and  in-circles  for  n  =  4,  8,  16,  32,  given  R  =  10. 

47.  Repeat  example  45  for  n  ==  6,  12,  24,  48. 

48.  Repeat  example  46  for  n  =  6,  12,  24,  48. 

49.  A  body  is  acted  upon  by  three  forces  of  magnitudes  20,  40,  60,  parallel 
to  the  sides  of  an  equilateral  triangle.  Resolve  these  forces  along  two  perpen- 
dicular axes,  then  combine,  and  thus  find  the  magnitude  and  direction  of  the 
resultant. 

50.  A  body  situated  at  one  vertex  of  a  regular  hexagon  is  acted  upon  by 
five  forces  represented  in  magnitude  and  direction  by  the  vectors  drawn  to  the 
five  other  vertices.  Resolve  along  and  perpendicular  to  the  diameter  through 
the  point  and  find  the  magnitude  and  direction  of  the  resultant. 

51.  A  point  describes  a  circle  with  uniform  speed.  Determine  the  position 
of  its  projection  upon  a  diameter  in  terms  of  its  angular  displacement  from  that 
diameter. 

52.  A  point  describes  a  circle  of  radius  30  feet  at  a  rate  of  4  revolutions 
per  minute.  Find  the  position  of  its  projection  upon  a  diameter  at  the  end  of 
5  seconds  after  passing  the  extremity  of  that  diameter. 

53.  Determine  the  components  of  the  vertical  acceleration  g  along  and 
perpendicular  to  a  plane  inclined  at  an  angle  a  to  the  horizon. 

54.  If  ^r  =  32,  find  the  acceleration  along  and  perpendicular  to  a  plane 
whose  inclination  to  the  horizontal  is  30°,  15°,  10°,  5°. 

55.  A  man  weighing  150  pounds  stands  midway  on  a  30-foot  ladder  whose 
foot  is  10  feet  horizontally  from  the  vertical  wall  against  which  it  leans. 
Find  the  normal  (perpendicular)  pressure  on  the  ladder  and  the  force  tending 
to  cause  him  to  slide  along  the  ladder. 

56.  Find  the  components  along  the  X-  and  F-axes  of  a  force  of  65  pounds 
making  an  angle  of  28°  13'  with  the  Z-axis. 

57.  A  steamer  is  sailing  in  such  a  way  that  its  speed  due  east  is  12  miles 
per  hour  and  its  speed  due  south  is  14  miles  per  hour.  Find  the  direction  of 
the  steamer's  course  and  the  speed  in  that  course. 

58.  In  an  oblique  triangle,  angle  B  =  45°,  angle  C  =  32°,  and  side  b  =  16. 
Find  side  c.  (Suggestion.  Draw  the  perpendicular  from  the  vertex  A  upon 
the  opposite  side.)  Attempt  to  deduce  a  general  relation  between  the  func- 
tions of  the  acute  angles  of  au  oblique  triangle  and  the  opposite  sides. 


CHAPTER   IV 

LOGARITHMS 
21.    Definition  of  a  logarithm.     If  we  have  given 

we  can  find  the  product  of  5Q  and  79  without  performing  the 
operation  of  multiplication,  provided  we  know  in  advance  the 
powers  of  10.  For,  we  have  from  the  general  laws  governing 
exponents, 

56  X  79  =  10i-^4«i^  X  W'^''^' 

__  -j^Ql.74819+1.89763 
=:103-«4'^82^4424. 

It  will  be  seen  that  the  process  of  multiplication  has  been  replaced 
by  the  simpler  one  of  addition. 

Many  other  processes  in  computation  can  be  simplified  in  a 
similar  manner ;  for  example,  if  we  wish  to  find  the  cube  root  of 
a  number,  say  89.1,  we  have 

89.1  =  101-94988^ 

and  consequently  • 


V89.1  =  (10i-94988y  =  100-64996  ^  4.466+. 

In  this  case  the  extraction  of  a  root  has  been  accomplished  by  the 
simple  process  of  division.  In  order  to  extend  this  method  we 
must  know  all  of  the  powers  of  some  convenient  number.  The 
exponents  involved  are  called  logarithms,  and  the  number  raised 
to  a  power  is  referred  to  as  the  base  of  the  logarithmic  system. 
We  may  define  a  logarithm  more  exactly  as  follows : 

If  a  is  any  number  and  x  and  n  are  so  related  that  «^  =  n^  then 
X  is^,«^lled  the  logarithm  of  n  to  the  base  a ;  that  is,  a  logarithm  is 
the  index  of  the  power  to  which  the  base  must  be  raised  to  obtain 
the  given  number. 

This  relation  is  denoted  symbolically  by  writing 

X  =  log„  n, 

and  is  read  ^'•x  is  equal  to  the  logarithm  of  n  to  the  base  a." 

29 


30  LOGARITHMS 

Thus  3  is  the  logarithm  of  8  to  the  base  2,  since  2^  =  8  ;  and 
in  the  illustrations  given  above,  1.74819  is  the  logarithm  of  66  to 
the  base  10,  etc. 

The  two  statements 

a^  =  n,  x  =  logo  n 

are  inverse  to   each  other,  just  as  are  the  relations  sin  x  and 
arcsin  x^  etc.,  of  Art.  14. 

Exercise.  Find  by  inspection  log3  27,  log5.625,  log8  32, 
log,. 04.  .        .        .         , 

The  logarithm  of  a  number  to  itself  as  base  is  unity,  since  71^  =  71. 

The  logarithm  of  1  to  any  base  other  than  zero  is  zero,  since 
a^  =  1. 

In  conformity  with  the  definition  just  laid  down,  it  follows  that, 
if  two  numbers  are  equal,  their  logarithms  to  the  same  base  are 
equal.  It  is  also  true  conversely,  that  if  the  logarithms  of  two 
numbers  to  the  same  base  are  equal,  the  numbers  are  equal.* 

If  the  base  is  real  and  positive,  real  logarithms  produce  only 
positive  numbers.  If  the  base  is  real  and  negative,  even  loga- 
rithms produce  positive  numbers  ;  odd  logarithms,  negative  num- 
bers. For  this  reason  only  real  positive  bases  are  chosen  in  prac- 
tice, and  only  positive  numbers  are  combined  by  the  aid  of  their 
logarithms.  The  sign  of  the  result  is  ascertained  entirely  apart 
from  the  numerical  computation. 

22.  Laws  of  combination.  Logarithms  are  important  in  trigo- 
nometry and  elsewhere  as  labor-saving  devices  in  calculations  with 
numbers  containing  many  digits.  Only  so  much  of  the  theory  of 
logarithms  as  is  necessary  for  this  purpose  will  be  developed  in  the 
present  chapter. 

The  laws  of  combination  of  numbers  by  the  aid  of  their  loga- 
rithms follow  at  once  from  the  definition  of  the  preceding 
article. 

I.  The  logarithm  of  the  product  of  two  factors  is  equal  to  the  sum 
of  their  logarithms,  all  to  the  same  hase. 

For,  if  a;  =  log„  n  and  y  =  log^  m  we  may  write  • 

n  =  a^  and  m  =  a^. 

*  In  the  theory  of  analytic  functions  a  broader  definition  of  the  logarithm  is  laid 
down,  and  the  statement  just  made  requires  modification. 


LAWS   OF  COMBINATION  31 

Multiplying,  we  have,  by  the  exponential  law, 

nm  =  a^+^, 

whence,  loga  nm  ^x-\-y=  logc*  ^  +  lo&a  ^-  (1 ) 

This  law  may  evidently  be  extended  to  any  finite  number  of 
factors. 

II.  The  logarithm  of  the  quotient  is  equal  to  the  logaritJim  of  the 
dividend  minus  the  logarithm  of  the  divisor^  all  to  the  same  base. 

For,  if  a;  =  log^ri  and  y  =  log^  m,  we  may  write  as  before, 

n  =  a*,  m  =  a^. 

n 
Dividing,  we  have  —  =  a^"^, 
m 

whence,  log„  ^— j  =x  —  y  =  log«  n—  log„  m.  (2) 

Manifestly  log«  f  —  j  =  —  log^  m. 

III.  The  logarithm  of  the  power  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  multiplied  by  the  index  of  the  power. 

For,  if  a;  =  log^  n,  then  n  =  a^. 

Hence,  n^  =  {a^y  =  a^"^ 

or,  log«  (nP)  =px  =  p  loga  n-  (3) 

IV.  The  logarithm  of  the  root  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  divided  by  the  index  of  the  root. 

For,  if  :r  =  log^  n,  then  n  =  a^.  Extracting  the  ^'th  root  of  both 
members,  we  get 

v  n  =  aQ, 

whence,  log^^^  =  -  =  i  log„  n.  (4) 

q      <1 

23.  Common  logarithms.  Any  number  may  be  used  as  a  base 
of  a  system  of  logarithms.  For  certain  purposes  the  so-called 
natural  system  of  logarithms,  which  has  for  its  base  the  number 
e  =  2.71828183  •••,  has  advantages.  For  the  purposes  of  ordinary 
numerical  computation,  however,  it  is  most  convenient  to  employ 
for  the  base  of  the  system  of  logarithms,  10,  the  base  of  the 
universally  adopted  system  of  numeration. 


32  LOGARITHMS 

The  common  logarithms  of  all  exact  integral  powers  of  10  are 
positive  integers ;  for  instance 

logio  (1000000)  =  logio(W) 
=  6  log,,  10 
=  6. 

The  logarithms  of  reciprocals  of  integral  powers  of  10  are 
negative  integers ;   thus 

logio  (.00001)  =logi„  (10-0 
=  -51og,„10 
=  -5. 

The  losrarithms  of  numbers  situated  between  two  consecutive 
integral  powers  of  10,  say  between  10^  and  10*+^,  lie  between  k 
and  k  +  1,  where  k  is  any  integer,  positive  or  negative.     Thus 

103  <  2417  <  104, 

whence,  ,  3  <  log^^  2417  <  4, 

or,  logjQ  2417  =  3  +  a  number  lying  between  0  and  1. 

The  logarithms  of  numbers  greater  than  the  base  consist  of  an 
integer  plus  a  proper  fraction.  The  fractional  part  is  written 
decimally,  calculated  to  a  number  of  decimal  places,  depending 
on  the  degree  of  accuracy  desired  in  the  use  of  the  table.  The 
integral  part  of  the  logarithm  is  called  the  characteristic;  the 
decimal  fraction,  its  mantissa. 

Hereafter,  in  this  book,  except  in  Chapter  IX,  we  shall  have 
to  do  only  with  common  logarithms  and,  unless  otherwise  expressly 
stated,  log  n  will  denote  logjQ  n. 

24.  Characteristic.  If  one  number  is  equal  to  another  number 
multiplied  by  a  factor  which  is  a  power  of  10,  the  logarithms  of 
the  two  numbers  differ  by  an  integer.     For 

log  (10*  xn')=  log  (10^  +  log  n 

=  k  +  log  n. 

Example.  log  34000  =  3  +  log  34 

=  4  +  log  3.4,  etc. 

Every  number  containing  one  digit  at  the  left  of  the  decimal 
point  lies  between  10^  and  10^.     The  characteristic  of  its  logarithm 


CHARACTERISTIC.    MANTISSA  33 

is  therefore  0.  The  cipher  should  always  be  written  to  indicate 
that  the  characteristic  has  not  been  overlooked. 

Every  number  containing  k  digits  at  the  left  of  the  decimal 
point  is  10*"^  times  a  number  with  one  digit  at  the  left.  The 
characteristic  is  therefore  k  —  \.  We  have  then  the  following 
rule  for  the  characteristic  : 

The  characteristic  of  the  logarithm  of  any  number  greater  than 
unity  is  one  less  tlian  the  number  of  digits  at  the  left  of  the  decimal 
point. 

Should  the  number  be  less  than  unity,  move  the  decimal  point 
ten  places  to  the  right  (thus  multiplying  by  10^^)  and  apply  the 
same  rule  as  before,  then  write  —  10  after  the  logarithm  for 
correction.     Thus 

log  7.12  =  0.85248, 

log  71200  =  log  (10*  X  7.12) 

=  4.85248, 
log  .00712  =  log  (10-10  X  71200000) 
=  log  (10-10  X  107  X  7.12) 

=  7.85248-10. 

The  positive  part  of  the  last  characteristic  is  seen  to  be  the 
difference  found  by  subtracting  from  9  the  number  of  ciphers 
immediately  following  the  decimal  point  in  the  number. 

The  characteristic  of  the  logarithm  of  any  number  less  than  unity 
is  found  by  subtracting  from  9  the  number  of  ciphers  between  the 
decimal  point  and  the  first  significant  digit,  then  affixing  —10. 

25.  Mantissa.  We  have  seen  that  moving  the  decimal  point 
in  the  number  merely  changes  the  characteristic  of  the  logarithm, 
leaving  its  mantissa  unaltered.  The  mantissa  depends  solely  upon 
the  sequence  of  significant  digits. 

In  the  tables  given,  the  logarithms  are  computed  to  five  deci- 
mal places  (see  pp.  1-21),  and  the  mantissas  alone  for  all  numbers 
from  100  to  9999  are  given,  arranged  in  the  following  manner : 
Running  down  the  left  margin,  under  iV,  are  to  be  found  the 
first  three  digits  of  the  number.  In  the  next,  (open)  column 
occur  the  first  two  figures  of  the  mantissa.  In  the  next  ten 
columns  are  the  remaining  three  figures  of  the  mantissa  arranged 
under  the  fourth  digit  of  the  number  at  the  top  of  the  columns. 


34  •  LOGARITHMS 

Thus  to  find  the  mantissa  of  log  3814,  we  select  the  row  having 
381  in  the  left  margin.  The  first  two  figures  of  the  mantissa,  58, 
are  found  in  the  first  column.  The  three  remaining  figures,  138, 
are  found  in  the  column  headed  4,  the  fourth  digit  of  the  number, 
giving  the  mantissa  .58138. 

To  avoid  repetition,  the  first  two  figures,  58,  are  not  printed  in 
every  line,  but  are  to  be  used  from  3802  to  3890,  inclusive.  The 
prefixed  asterisk,  *006,  denotes  that  the  mantissa  of  3891  is  .59006, 
not  .58006. 

EXERCISE   IX 

1.  Find  by  inspection  logg  16,  log,,  27,  log^  ^^. 

2.  Find  by  inspection  logg  81,  logg  32,  logo;  9. 

3.  What  numbers  correspond  to  the  following  logarithms  to  base  4  :  0,  1, 
2,2.5,3,  -2,-3? 

4.  What  numbers  correspond  to  the  following  logarithms  to  base  8  :  0, 
1,H,  -I,  -2? 

5.  Find  by  logarithms:  («)^;  (P)  '^^^f^iM. 

6.  Find  («)  VtW;  (b)  \/W7 ;  (c)  \/9l. 


^   Find^ 


18  X  V240  X  753 
72  X  Vim  X  200 

3/ 


,   Find  (^xV720xl5Y^ 
V2x  V480x  248/ 


9.   Find  { "^^  ]  *  ,  where  k  =  1.41. 


\  65  / 


10.  Solve  for  x:  ^^  =  24. 

11.  Solve  for  x :  6*  =  25. 

The  amount  A  attained  by  a  principal  P  at  interest  at  the  rate  r  com- 
pounded annually  for  n  years  is 

A  =P(1  +r)~. 

12.  Find  the  amount  of  $  3680  at  4  per  cent  in  6  years. 

13.  Find  the  principal  which,  in  7  years  at  5  per  cent,  amounts  to  ^  5820. 

14.  At  what  rate  will  ^  5000  amount  to  $7500  in  8  years  ? 

15.  In  how  many  years  will  $  86,500  amount  to  $  129,600  at  3^  per  cent  ? 

16.  If  a  city  increases  its  population  I  each  year,  in  how  many  years  will 
it  double  its  size  ? 


INTERPOLATION  35 

26.  Interpolation.  It  will  be  shown  in  Art.  79  that  the  differ- 
ence in  the  logarithms  of  two  numbers  is  approximately  propor- 
tional to  the  difference  in  the  numbers  provided  these  differ- 
ences are  small.     Thus,  approximately, 

log  51473  -  log  51470  ^  51473  -  51470  ^  3 
log  51480  -  log  51470      51480-51470      10* 

We  have,  then, 

log  51473  =  log  51470  +  ^\  (log  51480  -  log  51470). 

Introducing  the  values  from  Table  I, 

log  51473  =  4.71155  +  ^^^  (4. 71164  -  4.71155) 

=  4.71155  4-. 3  X  .00009 

=  4. 71155 +.00003 

=  4.71158. 

The  difference  .00009,  or  omitting  the  denominator,  the  9  is 
called  the  tabular  difference  corresponding  to  the  logarithm  of 
5147.  Note  that  the  added  difference  is  computed  to  the  nearest 
fifth  decimal  place. 

This  process  is  called  interpolation  by  the  principle  of  pro- 
portional parts.  To  facilitate  interpolation,  tables  of  proportional 
parts  are  inserted  in  the  logarithmic  tables  in  the  column  headed 
P.P.  At  the  top  of  each  of  the  P.P.  tables  is  the  tabular  differ- 
ence and  under  this  is  the  number  to  be  added  corresponding  to 
the  digit  at  the  left.     For  example 

log  38.25  =  1.58263 
log  38.26  =  1.58274. 

The  difference  is  .00011  and  in  the  P.P.  column  is  a  table 
headed  11.  Suppose  now  that  log  38.257  is  required.  Opposite 
7  under  11  is  found  7.7  ;  hence  8  is  to  be  added  in  the  fifth  deci- 
mal place,  giving 

log  38.257  =  1.58271. 

27.  Numbers  from  given  logarithms.  The  inverse  process  of 
finding  the  number  corresponding  to  a  given  logarithm  is  best 
explained  by  an  illustration.  Given  the  logarithm  3.84235.  Only 
the  mantissa  need  be  considered  at  first,  as  the  characteristic 
merely  determines  the  position  of  the  decimal  point  in  the  number. 


36  LOGARITHMS 

Looking  for  84  in  the  first  column  after  the  margin,  we  find  it 
corresponding  to  numbers  from  692  to  707.  The  nearest  tabular 
number  (mantissa)  smaller  than  235  is  230,  corresponding  to  the 
number  6955.  The  difference  is  5,  while  the  tabular  difference, 
found  by  subtracting  230  from  236,  is  6.  We  have  now  the  pro- 
portion for  the  next  digit, 

n  _5  ^ 

10~6' 

so  that  the  next  digit  is  found  by  dividing  50  by  6.  It  is  inad- 
visable to  carry  the  interpolation  beyond  one  additional  digit. 
Since  50  -^  6  =  8  •  +  •  •  •,  we  have  found  the  desired  number  to  be 
6955.8.  The  decimal  point  is  placed  after  the  fourth  digit  accord- 
ing to  the  rule  for  the  characteristic,  the  given  characteristic 
being  3.  Should  the  logarithm  be  followed  by  — 10,  the  decimal 
point  must  finally  be  moved  ten  places  to  the  left. 

28.  Cologarithms.  The  logarithms  of  divisors  have  to  be  sub- 
tracted. Subtraction,  however,  can  be  avoided  and  the  logarith- 
mic computation  of  a  succession  of  multiplications  and  divisions 
effected  by  a  single  addition  process.  There  is  no  advantage  in 
using  cologarithms  when  but  two  factors  are  involved.  When, 
however,  more  than  two  are  involved,  instead  of  dividing  by  the 
denominator  or  divisor  factors,  we  may  multiply  by  their  recipro- 
cals, obviously  a  legitimate  substitution.     Now 

log—  =  —  log  m  =  (10  —  log  m)  —  10. 
m 

This  logarithm,  (10  —  log  m)  —  10,  is  called  the  cologarithm 
of  m,  written  cologm.  It  may  be  written  down  immediately  from 
the  table  by  beginning  at  the  left  and  subtracting  each  figure  from 
9,  until  the  last  figure,  which  must  be  subtracted  from  10.     Thus 

log  28.24  =  1.45086 

and  colog  28.24  =  8.54914 -10. 

29.  Logarithms  of  trigonometric  functions.  Logarithms  of  the 
trigonometric  functions  are  arranged  in  Table  II  in  the  same 
manner  as  are  the  natural  functions,  or  true  numerical  values  of 
the  functions.  Logarithmic  secants  and  cosecants  need  not  be 
printed,  since  they  are  the  cologarithms  of  the  cosines  and  sines. 

The  sines  and  cosines  of  angles  and  the  tangents  of  angles  less 


LOGARITHMS   OF   TRIGONOMETRIC   FUNCTIONS  37 

than  45°  are  numerically  less  than  unity.  In  conformity  with 
Art.  24,  therefore,  their  logarithms  are  written  in  the  augmented 

^^^^^'  log  sin  6d°  21^  =  9. 95850  -  10. 

The  — 10  is  not  printed  in  the  table  but  it  is  always  understood. 
The  positive  portion  of  the  characteristic  is  printed  in  the  table. 
Usage  differs  with  respect  to  printing  the  logarithmic  tangents 
of  angles  greater  than  45°.  Engineering  and  physical  instruments 
are  usually  graduated  to  minutes  or  larger  divisions  of  the  angle, 
so  that  it  is  not  feasible  to  carry  the  interpolation  farther  than  to 
tenths  of  minutes.  The  tables  of  functions  and  of  proportional 
parts  printed  in  connection  with  this  book  are  arranged  with  this 
in  view. 

Astronomic  observations  justify  carrying  the  interpolation  to 
seconds,  and  astronomers  use  for  this  purpose  tables  computed  to 
seven  or  more  decimal  places. 

For  example, 

log  sin  29°  37'  =  9.69890  -  10, 

log  sin  29°  38'  =  9. 69412  -  10. 

The  difference  is  .00022,  and  in  the  P.P.  column  is  a  table  headed 
22.  Suppose  now  that  log  sin  29°  37.4'  is  required.  Opposite  4 
under  22  is  found  8.8 ;  hence  9  is  to  be  added  in  the  fifth  decimal 
place,  giving 

log  sin  39°  37.4'  =  9.69399  -  10. 

EXERCISE  X 

1.  Find  from  the  table  the  logarithms  of  72484,  619.25,  695  x  10^ 
.00064375,  3  x  lO^i. 

2.  Find  from  the  table  the  logarithms  of  91386,  14.295,  321  x  10^, 
.000078541,  2  x  lO^*. 

3.  Find  the  numbers  whose  logarithms  are  3.71295,  12.61242,  8.21312  -  10. 

4.  Find  the  numbers  whose  logarithms  are  4.21382,  11.75153,  6.13579  -  10. 

5.  Find  Young's  modulus  of  elasticity  from  the  formula  Y= — ^,  if 
m  =  4932.5,  g  =  980,  I  =  110.5,  tt  =  3.1416,  r  =  .25,  s  =  .3.  '^''  ^ 

6.  Find  the  radius  of  the  sun  if  its  mass  is  2.03  x  10^^  grams,  and  its 
average  density  is  1.41,  knowing  that  mass  =  volume  x  density. 

7.  The  radius  r  of  each  of  two  equal,  tangent,  iron  spheres  which  attract 
each  other  with  a  force  of  1  gram's  weight,  is  given  by  the  formula 

4r2  i22' 


38  LOGARITHMS 

in  which  the  density  of  iron  p  =  7.5,  the  mass  of  the  earth  ikf  =  6.14  x  10*^ 
grams,  and  the  radius  of  the  earth  R  —  6.37  x  10^  cm.,  while  ir  =  3.1416.  Solve 
for  r  and  compute  by  logarithms. 

8.  Solve  example  7  for  spheres  of  lead  with  density  p  =  11.3. 

9.  The  population  of  a  county  increases  each  year  by  12.5  per  cent  of  the 
number  at  the  beginning  of  the  year.  If  its  population  Jan,  1,  1776,  was 
2.5  X  106,  what  will  it  be  Dec.  31,  1926? 

10.  If  the  number  of  births  and  deaths  per  annum  are  3.5  per  cent  and 
1.2  per  cent  respectively  of  the  population  at  the  beginning  of  each  year,  and* 
the  population  on  Jan.  1,  1830,  was  5  x  10^  find  the  population  Jan.  1,  1905. 

11.  Find  from  the  tables  log  sin  25°  32.3',  log  cot  71°  18.6',  colog  cos  16° 
29.2'. 

12.  Find  from  the  tables  log  cos  19°  25.7',  log  tan  31°  16.2',  colog  sin 
65°  12.8'. 

13.  Find  the  angles  corresponding  to  log  cos  a  =  9.31723,  log  cot  y8  =  9.16251, 
log  tan  y  =  0.61253. 

14.  Find  the  angles  corresponding  to  log  sin  a  =  9.63152,  log  tati 
(3  =  9.71728,  log  cot  y  =  0.15382. 

15.  Francis  deduces  the  following  formula  for  the  discharge  over  a  weir, 
q  =  3.01  bH^-^,  in  which  q  is  the  discharge  in  cubic  feet  per  second,  b  the  breadth 
of  the  crest,  and  H  the  head  of  water.  Find  by  logarithms  the  discharge  when 
6  =  3.5  and// =1.2. 

16.  A  common  formula  for  finding  the  diameter  of  a  water  pipe  is 


m 


d  =  0.479 

h 

in  which /is  a  friction  factor,  I  the  length  of  the  pipe,  q  the  discharge,  and  h 
the  head.     Find  d  when  /=  0.02,  I  =  500,  ^  =  5,  ^  =  10. 

17.  The  discharge  from  a  triangular  weir  is  given  as  ^  =  c  I'V  V2  g  H^,  in 
which  c  is  a  constant,  g  the  acceleration  of  gravity,  and  H  the  head.  Find  q 
when  g  =  32.2,  H  =  1.2,  c  =  0.592. 

18.  The  formula  for  velocity  head  is  h  =  0.01555  V^.     Find  ?i  when  V  =  b. 

19.  The  elevation  of  the  outer  rail  on  what  is  known  as  a  one-degree  railwav 
curve  to  resist  centrifugal  force  is  sometimes  given  by  the  formula  e  =  0.00066  V^, 
e  being  in  inches  and  V  the  speed  of  the  train  in  miles  per  hour.  When  F  =  45, 
comjmte  e. 

20.  Another  expression  for  the  relation   of  the  preceding   problem  is 

qY-2 

c  =    '^        •     Here  e  is  in  feet,  g  is  the  gauge  of  the  track,  V  is  the  speed  in  feet 

per  second,  and  R  is  the  radius  of  the  curve.     Given  g  =  4.71,  V  =  66,  R  =  5730, 
compute  e. 


THE   SLIDE   RULE  39 

21.  The  difference  between  the  base  and  hypotenuse  of  a  right  triangle  is 

given  by  c  —  a  = ,  and  when  a  and  c  are  nearly  equal,  approximately  by 

yi  c  -\-  a 

c  —  a  =  — . 

2c 

Find  the  per  cent  of  error  introduced  by  the  second  method  when  the  angle 
between  a  and  c  is  15°. 

22.  If  a  =  length  of  a  short  circular  arc  and  c  =  its  chord,  then  approxi- 
mately a  —  c  =  .  Given  a  =  ^  and  R  —  100,  compute  the  value  of  this 
difference. 

23.  The  relation  between  the  pressure  and  volume  of  air  expanding  under 
certain  conditions  is  pj??j  ^-^^  —  pv^*\  where  p^  and  v^  are  initial  values.  If  p^  =  40, 
v^  =  5.5,  find  V  when  j9  =  24  ;  also  when  p  =  16. 

24.  The  relation  between  the  initial  and  final  temperatures  and  pressures 
is  given  by  the  equation 

With  ^j  =  60  and  the  other  data  as  in  Ex.  23,  find  the  final  temperatures 
for  p  =  24:  and  j9  =  16,  respectively. 

30.  The  slide  rule.  The  principles  of  logarithmic  computation 
are  conveniently  illustrated  by  means  of  the  slide  rule,  now  widely 
used  in  performing  mechanically  such  operations  as  admit  of  the 
use  of  logarithms.  A  brief  description  of  this  instrument  will  be 
found  profitable  at  th^s  stage,  and  its  use  by  the  student  as  a 
ready  check  upon  the  numerical  solution  of  problems  is  strongly 
recommended.  As  will  be  seen  by  an  inspection  of  the  simplified 
diagram  of  Fig.  19,  the  rule  is  essentially  a  device  for  adding  and 


A  B    Rule  C 

1  2  3  4  5         6      7 


a      Slide  h 


1 1 — I — I — I — 

3  4  5         6789    10 


Fig.  19. 

subtracting  logarithms,  thereby  giving  a  wide  range  of  computa- 
tions. In  the  figure  the  point  6X  on  the  "slide"  is  set  opposite 
the  point  B  on  the  "rule."  If  both  scales,  which  are  alike,  are 
so  divided  that  AB  is  equal,  or  proportional,  to  log  2  and  ab  to 
log  3,  then  0  on  the  rule  opposite  b  on  the  slide  gives  the  distance 
^(7  equal,  or  proportional,  to  log  6.  That  is,  log  2  +  log  3  =  log 
(2x3)  =  log  6. 

Similarly  by  subtraction,  AC  —  ab  =  AB ; 

that  is  log  6  —  log  3  =  log  2. 


40 


LOGARITHMS 


The  point  a  of  the  slide  is  called  the  index^  hence  we  have  the 
following  rules  for  simple  operations. 

1.  To  multiply  two  numbers,  set  the  index  opposite  one  num- 
ber on  the  rule  and  opposite  the  other  number  on  the  slide  read 
the  product  on  the  rule. 

2.  To  divide  one  number  by  another,  set  the  divisor  on  the 
slide  opposite  the  dividend  on  the  rule  and  read  the  quotient  on 
the  rule  opposite  the  index. 

In  the  instrument  as  actually  constructed,  *  Fig.  20,  there  are  four  scales 
denoted  respectively  by  A,  B,  C,  and  D,  of  which  scales  B  and  C  are  on  the 


Fig.  20. 

slide.  For  convenience  in  compound  operations  the  rule  is  provided  with  a 
runner  r  by  means  of  which  a  setting  of  the  slide  may  be  preserved  while  the 
slide  is  moved  to  a  new  position.  The  following  example  will  illustrate  the 
manipulation  of  slide  and  runner. 

Example  1.     Find     6^^^115x27. 
14.6  X  342 

Set  14.6  on  C  scale  opposite  63  on  D  scale ;  move  runner  to  115  on  C  scale ; 
move  342  on  C  scale  to  runner,  and  opposite  27  on  C  scale  read  result  on  D  scale. 

In  this,  as  in  all  slide-rule  computations,  the  decimal  point  must  be 
located  by  inspection. 

On  the  lower  side  of  the  slide  are  three  scales,  the  outer  of  which  are  marked 
S  and  Irrespectively.     The  following  examples  illustrate  the  use  of  these  scales. 

Example  2.     Find  36  sin  22^ 

Set  22  on  the  S  scale  opposite  the  mark  on  the  slot  in  the  right-end  of  the 
rule ;  then  opposite  the  end  of  the  A  scale  can  be  read  on  the  B  scale  the  natu- 
ral sine  of  22°.     Now  opposite  36  on  the  A  scale  read  the  result  on  the  B  scale. 

Example  3.     Find  26.5  tan  13°  15'. 

Reverse  the  slide  and  set  13°  15'  on  the  T"  scale  opposite  the  mark  on  the 
slot ;  then  opposite  the  end  of  the  B  scale  can  be  read  on  the  D  scale  the  natu- 
ral tangent  of  13°  15'.  Set  the  runner  at  this  point  and  replace  the  slide  with 
the  index  at  the  runner.  Opposite  26.5  on  the  C  scale  read  the  required  prod- 
uct on  the  D  scale. 

Example  4.     Find  56i''. 

Set  the  index  of  C  scale  opposite  56  on  D  scale  and  opposite  the  mark  on 
the  under  side  of  the  right-hand  end  of  the  rule  read  748  on  the  middle  scale 

*  A  more  detailed  description  of  the  slide  rule  is  not  within  the  scope  of  this 
book.  A  manual  describing  fully  the  use  of  tlie  instrument  can  be  had  of  any  firm 
selling  slide  rules. 


LOGARITHMIC   SOLUTION  OF   RIGHT   TRIANGLES  41 

of  the  lower  side  of  the  slide.  This  reading  is  the  mantissa  of  the  logarithm 
of  56.  The  characteristic  1  must  be  supplied  as  usual.  Now  in  the  usual  way- 
find  1.3  X  1.748;  that  is,  put  index  to  1.748  on  D  scale  and  opposite  1.3  on  C 
scale  read  the  product  2.272.  This  is  the  logarithm  of  SG^-^.  Set  the  mantissa 
272  on  the  logarithm  scale  opposite  the  mark  on  the  rule  and  read  118.7  on  the 
D  scale  opposite  the  index. 

EXERCISE  XI 

1     17-  /I  ^  \  64  X  37      ,,.       193  .  .   0.05  x  137  x  62 

1.  liind  (a)  (b)   — ;     (c) • 

^  163  ^  ^  67  X  2.1      ^  ^     14  X  28  X  6.5 

2.  Find  (a)  127  sin  24°,  (6)  0.32  sin  72°,  (c)  16.5  cos  35°. 

3.  Find  (a)  37  tan  8°  20',  {h)  1.35  tan  40°  10'. 

4.  Find  («)    11^2^32:,     (/.)  35.5  ?H^^ 

^  ^  64  ^  ^  sin  47° 

5.  Find  (a)  28^  {h)   y/^^,  (c)  7.311-27. 

31.  Right  triangles  solved  by  logarithms.  — It  is  now  possible, 
with  the  aid  of  the  logarithmic  tables,  to  solve  right  triangles  the 
numerical  values  of  whose  parts  contain  more  digits  than  those 
given  in  Chapter  III,  without  entailing  laborious  multiplications 
and  divisions. 

Example  1.     Given  a  =  51.72,  jS  =  73°  46^ 

Solving  the  proper  formulas  for  the  unknown  parts,  we  have 

a 

^  = B' 

cos  p 

h=a tan  /3, 

A  =  \a^  tan  ^, 

h  —  c  cos  a,  check. 

Sum  of  angles  =90°  00^ 

^=73°  46^ 

«=16°14^ 

log  «=  1.71366 

log  cos /3=  9.44646 -10 

log  (?  =  2.26720 

.•.c=  185.01 


42  LOGARITHMS 

log  a  =  1.71366 
log  tan  13  =  0.53587 
log  6  =  2.24953 
.-.  5  =  177.64 


21og«=    3.42732 
log  tan /3=    0.53587 
colog  2  =   9.69897-10 
log  ^  =  13.66216 -10 
.-.  J.  =  4593.67 

Check 
logc=    2.26720 
log  cos  a  =    9.98233-10 
log  5  =  12.24953 -10 
.-.5  =  177.64 

Note  that  log  a^=2  log  a.  In  solving,  first  write  all  the  forms 
needed  for  the  complete  solution  ;  secondly,  look  up  and  write  in 
all  the  needed  logarithms  of  the  data  from  the  tables ;  thirdly,  per- 
form the  additions  and  subtractions  ;  lastly,  from  the  logarithmic 
results  find  the  numbers.  Then  log  cos  /3,  log  tan  /3,  and  log 
cos  a  ( =  log  sin  )S)  can  all  be  found  from  once  turning  to  the 
angle  73°  46'. 

A  form  of  computation  sometimes  used  is  given  below.  It  has 
the  advantage  of  being  more  compact  than  the  usual  form,  and 
furthermore  the  logarithms  of  the  data  stand  close  to  the  data, 
thus  permitting  easy  verification  of  results  or  correction  of 
mistakes.  ^^^^^ 

a=51.72  log  1.71366  logl. 71366 

/3  =  73°  46'  log  cos  9.44646-10  log  tan  0.53587 


c  =  185.01 

log  2.26720 

2 
^  =  4593.7 

log  2.26720 

5  =  177.64 
«  =  16°14' 
5  =  177.64 

log  2.24953 

log  cos  9.98233- 10 
log  2.24953 

log  3.42732 
log  tan  0.53587 
colog  0.69897 -10 
log  3.66216 

LOGARITHMIC   SOLUTION  OF   RIGHT   TRIANGLES  43 

Example  2.    Given  5  =  7124.5,  c  =  9365.4. 
We  have, 


cos  a  = 

1 
c 

^  = 

:  90°  -  a. 

a  = 

■  e  sin  a, 

A  = 

:  ^  he  sin  a, 

a  = 

•■  h  tan  a,  check. 

log  5  = 

:  3.85275 

logc  = 

:  3.97153 

log  cos  a  = 

:  9.88122- 

10 

a  = 

:  40°  28.4^ 

0  = 

:  49°  31.6' 

loge^ 

:    3.97153 

log  sin  a  = 

:    9.81231- 

-10 

loga  = 

=  13.78384- 

-10 

a  = 

:  6079.2 

Check 

log  5  = 

:    3.85275 

log  tan  a  = 

:    9.93109- 

-10 

log  a  =  13.78384 -10 
a  =6079.2 

The  following  is  the  compact  arrangement  of  the  computation  : 

Check 
b  =  7124.5  log  3.85275  log  3.85275 

c  =  9365.4  log  3.97153  log  3.97153 

a  =  40°  28.4^      log  cos  9.88122-10   log  sin  9.81231- 10   log  tan  9.93109- 10 
13  =  49°  31.6^ 

a  =  6079.2  log  3.78384 

a  =  6079.2  log  3.78384 

It  appears  that  the  Pythagorean  proposition,  a^  +  5^  =  c^,  is 
not  used  because  it  is  not  adapted  to  the  use  of  logarithms.  It 
might  be  used  in  this  case,  however,  in  the  form 


44 


LOGARITHMS 


EXERCISE  XII 


Find  the  missing  parts  of  the  following  triangles,  using  loga- 
rithms.     (The  work  may  be  checked  with  a  slide  rule.) 


a 

fi 

a 

b 

c 

A 

1. 

63° 

2584 

2. 

7531 

8642 

3. 

75°  15.2' 

965.24 

4. 

47.193 

3972.6 

5. 

7.3298 

6.1032 

6. 

18°  25.5' 

32.96 

7. 

132.97 

985.27 

8. 

53.215 

13.712 

9. 

65983 

72916 

10. 

29°  50.2' 

10.207 

11. 

25°  17.4' 

382.97 

12. 

.00020 

.00037 

13. 

63°  12.7' 

7.1436 

14. 

.07154 

.09127 

15. 

35°  16.4' 

.62961 

16. 

35°  16.8' 

41658 

17. 

.00615 

.00415 

18. 

80°  12.5' 

5.2108 

19. 

.00729 

.01625 

20. 

25°  18.2' 

1729.3 

The  examples  1-20  of   Exercise  VIII  may  also  be  solved  by 
logarithms  and  the  results  compared  with  those  there  obtained. 

21.  Find  the  radius  of  the  circle  inscribed  in  a  regular  pentagon  whose 
side  is  12  feet. 

22.  Find  the  side  of  a  regular  pentagon  inscribed  in  a  circle  whose  radius 
is  15  feet  7  inches. 

23.  Find  the  area  of  a  regular  octagon  whose  circumscribed  circle  has  a 
diameter  of  10  feet. 

24.  A  tower  120  feet  high  throws  a  shadow  69.2  feet  long  upon  the  plane 
of  its  base.     What  is  the  angle  of  inclination  of  the  sun? 

25.  The  top  of  a  certain  lighthouse  is  known  to  be  73  feet  above  the 
water.  From  a  boat  the  angle  between  the  top  and  its  reflection  is  measured 
as  6°  45'.     How  far  is  the  boat  from  the  light  ? 

26.  Two  trains  leave  a  station  at  the  same  time,  one  going  north  at  the 
rate  of  30  miles  per  hour,  and  the  other  east  at  the  rate  of  40  miles  per  hour. 


PROBLEMS   INVOLVING  RIGHT   TRIANGLES 


45 


How  far  apart  will  they  be  in  20  minutes,  and  what  is  the  direction  of  the  line 
joining  them  ? 

27.  Show  that  if  a  is  the  side  of  a  regular  polygon  of  n  sides,  the  area  of 

1                   1  S()° 
the  polygon  is  given  by  ^4  =  -  d^n  cot 

28.  Show  that  if  r  is  the  radius  of  a  circle,  then  the  area  of  a  regular  cir- 

cumscribed  polygon  of  n  sides  is  A  =  rhi  tan 

n 

29.  Find  a  value  for  the  area  of  an  inscribed  polygon  corresponding  to 
that  given  above. 

30.  Taking  the  moon's  diameter  as  31'  20"  and  its  distance  from  the  earth 
as  239,000  miles,  what  is  its  diameter  in  miles  ? 

31.  At  what  distance  may  a  mountain  4  miles  high  be  seen  across  a  plain, 
the  earth  being  taken  as  a  sphere  of  4000  miles  radius? 

32.  If  the  sun's  diameter  is  taken  at  866,000  miles  and  its  distance  from 
the  earth  as  93,000,000  miles,  what  angle  should  it  subtend  at  the  center  of  the 
earth. 

33.  An  approximate  formula  for  the  distance  from  the  midpoint  of  a  cir- 
cular arc  to  the  midpoint  of  its  chord  is  m  =  -  — — — ■, 

4   100 

in  which  I  is  the  length  of  the  chord  in  feet  and  a  the 
deflection  or  circumferential  angle  subtended  by  a  base 
or  chord  of  100  feet.     Find  m  for  Z  =  30,  a  =  2°. 

34.  If  /  is  the  angle  of  intersection  between  two 
tangents  to  a  circle  of  radius  i2,  the  distance  T  from 
a  point  of  tangency  to  the  point  of  intersection  is  given 

hj  T  =  Rcod.     Find  T  ior  R  =  3000  feet,  and  /  =  22°  52'. 

35.  The  length  of  a  chord  is  given  by  2  72  sin  i  7,  in  which  /  is  the  central 
angle.     Find  the  chord  length  for  R  =  2000,  /  =  12°  13'. 

36.  A  river  which  obstructs  chaining  on  a  survey  is  passed  by  tri- 
angulation.  The  line  ^J5,'Fig.  22,  is  measured  200 
feet  perpendicular  to  AC,  and  the  angle  ADC  found 
to  be  35°  27'.     AVhat  is  the  distance  AC 2 

37.  With  an  instru- 
ment at  A,  Fig.  23,  a 
level  line  of  sight  passes 
6  ft.  above  the  top  of  a  wall  as  measured 
on  a  rod.     The  angles  of  depression  *  to 
the  top  and  bottom  of  the  vertical  face 
are    respectively,    2°    31'    and    42°    16'. 
What  is  the  height  of  the  wall? 
*  The  angles  of  elevation  and  depression  of  an  object  measure  respectively  its 
angular  distance  above  or  below  the  horizon  of  the  observer. 


Fig 


Fig.  22. 


Fig.  23. 


46 


LOGARITHMS 


AC 


DAB 


Fig.  24. 
14°  41'    find  AD  and  DB. 


38.  In  order  to  obtain  both  the  horizontal 
and  vertical  distances  to  an  inaccessible  point, 
the  solution  of  two  triangles  may  be  necessary. 
Fig.  24  represents  two  views  of  the  problem. 
Wishing  the  distances  AD  and  BD,  first  lay  out 
the  base  line  A  C  of  any  convenient  length  per- 
pendicular to  AB.  Measure  the  angle  A  CD  and 
compute  AD. 

Next  from  AD  and  the  angle  DAB^  the 
angles  of  elevation,  compute  DB. 

Having    ^C  =  300    ft.,    ACB  =  Ql°d4:',   and 


CHAPTER  V 


THE   OBTUSE   ANGLE 


32.  Definitions  of  the  trigonometric  functions  of  obtuse  angles. 
If  an  obtuse  angle  (^.e.  an  angle  greater  than  90°  and  less  than 
180°)  is  placed  on  the  axes  of  coordinates  in  the  same  manner  as 
was  the  acute  angle  in  Art.  6,  the  terminal  line  will  extend 
into  the  second  quadrant.  The  trigonometric  functions  of  such 
angles  are  defined  exactly  as  in  Art.  6.     Thus  in  Fig.  25, 


sin  a 


I 


cos  a 


tan  a 


,  etc. 


4F 


33.  Signs  and  limitations  in  value.  The  abscissas  of  all  points 
in  OA  (Fig.  25)  are  negative,  while  their  ordinates  and  radii 
vectores  are  positive.  It  is  evident,  therefore,  that  some  of  the 
defining  ratios  are  negative.  In 
accordance  with  the  law  of  signs 
in  algebraic  division,  we  find 
that  the  sines  and  cosecants  of  all 
obtuse  angles  are  positive,  while 
their  cosines,  secants,  tangents, 
and  cotangents  are  negative. 

The  student  should  verify 
each  of  these  statements  in  de- 
tail and  become  unhesitatingly 
familiar  with  these  fundamental 
facts. 

P\irthermore,  the  sine  and  cosine  cannot  be  numerically  greater 
than  unity  and  the  secant  and  cosecant  cannot  be  numerically 
less  than  unity. 

47 


^X 


Fig.  25. 


48 


THE   OBTUSE   ANGLE 


Query.  What  are  the  limitations  in  value  of  the  tangent  and  co- 
tangent ? 

34.  Fundamental  relations.  If  the  effects  of  the  law  of  signs 
are  traced,  it  will  be  seen  that  all  the  relations  of  Art.  9  hold  also 
for  functions  of  an  obtuse  angle  without  any  modifications. 

35.  Variation.  As  the  angle  6  varies  from  90°  to  180°,  while 
V  remains  constant,  x  is  always  negative  and  varies  from  0  to  —  v, 
and  y  is  positive  and  varies  from  v  to  0.  Consequently,  as  6 
increases  from  90°  to  180°,  sin  6  decreases  from  1  to  0,  cos  6  decreases 
(algebraically)  from  0  to  —  1,  tan  6  increases  from  —  oo  to  0,  cot  6 
decreases  from  0  to  —  oo,  sec  6  increases  from  —  oo  to  —  1,  esc  6 
increases  from  1  to  oo. 

The  terms  positive  infinity  and  negative  infinity  require  careful 
consideration.  If  6  varies  continuously  from  89°  to  90°,  tan  0 
varies  in  such  a  way  as  to  exceed  in  magnitude  any  previously 
assigned  definite  value,  however  large.  As  it  is  positive  for  all 
values  of  9  in  the  first  quadrant,  it  is  consequently  said  to  become 
positively  infinite  (+oo).  If  6  varies  continuously  from  91°  to 
90°,  tan  6  varies  so  as  to  exceed  numerically  any  previously 
assigned  definite  value.  As  it  is,  however,  always  negative  for 
values  of  6  in  the  second  quadrant,  it  is  said  to  become  negatively 
infinite  (-co).  The  plus  or  minus  sign  written  before  the  symbol 
00  merely  indicates  whether  the  trigonometric  function  increases 
numerically  without  limit  through  a  positive  or  a  negative  set  of 
values. 

36.  Functions  of  i8o°.  As  6  approaches  180°,  v  remaining 
constant,    x    approaches    —  v   and    y    approaches    0.     We   have, 

\Y  then, 

sin  180°  =  0, 

cos  180°  =  -  1, 

tan  180°  =  0, 

cot  180°  =  00, 

sec  180°  =  -  1, 

FIG.  26.  cscl80°=oc. 


37.    Functions  of  supplementary  angles.    Two  angles  are  called 
supplementary  if  their  sum  is  180°.     Thus,  in  Fig.  27,  a  and  fi  are 


FUNCTIONS   OF   (180^^  -  a)    AND   (90°+ a) 


49 


supplementary,  and  yS=180  — «,  a  being  acute.  The  triangles 
OMP  and  ONQ  are  similar,  but  ON  is  negative.  The  pairs  of 
corresponding  sides  are  v  and  v\  x  and  r^:^  y  and  yK     Hence  we  have 


sin  (180°  -  a)  =  sin  /3 
cos  (180°  -  a)  =  cos  yS  =  ^ 


v'       V 
v' 


sm  a. 


=  —  cos  a, 


tan  (180°  -  a)  =  tan  ^ 

Similarly : 

cot  (180°  -  a)  =  -  cot  a, 

sec  (180°  —  a)  =  —  sec  a, 
esc  (180°  —  a)  =  CSC  a. 

As  a  consequence  of  the 
relation  sin  (180°  —  «)  =  sin  a, 
two  values  exist  for  arcsin  m, 
the  one  acute,  the  other  obtuse, 
and  supplemental  to  each  other. 

Y 


y  _    y 


—  tan  a. 


Y 

P^^ 

"^-^^ 

V  ^ 

y' 

^'       J 

f3       3^ 

y 

I 

ST 

X' 

X 

M 

Fig.  27. 

In   case   m  =  1,   the   two  values   are 
identical. 


Fig.  28. 


QdERY.     Is  this  also  true  of   arccos  m, 
arctan  tw,  etc.  ? 

38.    Functions  of    (90°  + a).      In 

y  Fig.   28,  ^  =  90°  +  a,  a  being  acute. 

-^f-^^     The  triangles    OMP   and    ONQ  are 

similar,  but  the  pairs  of  homologous 

sides  are  v^and  v\  x  and  y,  y  and  a;', 

while  2:'  is  negative.     We  thus  obtain 


sin  (90°  4-  a)  =  sin  /8  =  ^  =  -  =  cos  a, 


cos  (90°  +  a)  =  cos  /8 


_^_      ^_ 


sma, 


tan  (90°  +  a)  =  tan  /3 


-  =  —  cot  a. 


50  THE   OBTUSE  ANGLE 

In  like  manner, 

cot  (90°  +  a)  =  -  tan  a, 

sec  (90°  +  a)  =  —  CSC  a, 

esc  (90°  +  a)=  sec  a. 

EXERCISE    XIII 

1.  Find  the  values  of  the  functions  of  135°.     (See  Art.  11.) 

2.  Find  the  values  of  the  functions  of  150°.     (See  Art.  11.) 

3.  Find  the  value  of  sin  [cos-^(—  |f)],  tan  (csc-if  ^),  cos  [arctan  (—  :^)], 
the  angles  being  of  the  second  quadrant. 

4.  Find  the  value  of  cos  (arccos  —  ^j),  sin  [tan-i  (  —  j^^)],  cot  (arcsin  ff), 
the  angles  being  of  the  second  quadrant. 

5.  Express  in  terms  of  an  angle  less  than  45°,  cos  160°,  tan  130°,  sec  150°. 

6.  Express  in  terms  of  an  angle  less  than  45°,  sin  170°,  esc  95^  cot  140°. 

7.  Verify  for  a  =  60°,  the  ftquatiiuiig-  ^c/&  ^^^c/^S 

sin  2  a  =  2  sin  a  cos  a,  cos  2  a  =  2  cos^  a  —  1. 

8.  Verify  for  a  =  45°,  the  equations 

sin  3  a  =  3  sin  ct  —  4  sin^  a, 

cos  3  a  =  4  cos^  a  —  3  cos  a.  • 


9.   Verify  for  a  =  120°,  the  equations 


^^^  -  ^      -. ,-  +  cos  a 

cos        -'—'»' 


tanl«:=V^-"Q^^^^-^»^^. 
2  ^  1  +  cos  a         sin  a 

10.   Verify  for  a  =  120°,  the  equations 


i«=4 


cot-  ^--^1-  +  cos«_l  +  cosa 


cos  a         sm  a 

11.  Verify  for  a  =  120°,  (3  =  30°,  the  equations 

sin  («  +  ^)  =  sin  a  cos  /3  +  cos  a  sin  ^, 
cos  (a-  13)  =  cos  a  cos  )8  +  sin  a  sin  /S. 

12.  Verify  for  a  =  120°,  /3  =  60°,  the  equations 

sin  (a  —  )8)  =  sin  a  cos  /3  —  cos  a  sin  y8, 
cos  (a  -\-  f3)  =  cos  a  cos  /3  —  sin  a  sin  ^. 


FUNCTIONS   OF   OBTUSE   ANGLES  51 

13.   Fill  in  the  proper  values  in  the  following  table  for  handy  reference :" 


a 

sin  a 

cos  a 

tan  a 

cot  a 

sec  a 

CSC  a 

0° 

0 

30=^ 

i 

45° 

lV2 

60° 

W3 

90° 

1 

120° 

iV3 

135° 

iV2 

150° 

i 

180° 

0 

CHAPTER  VI 

OBLIQUE  TRIANGLES 

39.  Formulas  for  solution.  In  the  oblique  triangle  ABC, 
Fig.  29,  let  the  angles  be  denoted  by  a,  ^,  7,  and  the  lengths  of 
the  opposite  sides  by  a,  5,  c?,  as  in  the  figure. 

The  relation  a  4-  p  +  7  =  180°  al- 
ways exists,  and  consequently  when 
two  of  the  angles  are  known,  the 
third  is  determined.  Five  of  the  six 
parts  of  the  triangle  still  remain  to 
be  found ;  namely,  the  three  sides 
and  two  angles.  It  has  been  shown 
in  elementary  geometry  that  if  any 
three  independent  parts  are  given,  the  triangle  is  determined  and 
the  remaining  parts  can  be  found.  Then  two  formulas,  in  addi- 
tion to  the  one  just  stated,  are  sufficient  for  the  complete  solution. 
It  is,  nevertheless,  convenient  to  express  the  relations  between 
the  sides  and  angles  in  a  variety  of  forms.  Those  given  in  the 
following  pages  are  selected  on  the  score  of  utility.  They  fall 
into  sets  of  three  each.  From  any  one  of  each  set  the  other  two 
may  be  written  by  cyclic  advance  of  the  letters  involved  ;  i.e.  by 
changing  a  into  6,  h  into  c,  c  into  a,  and  at  the  same  time  a  into 
^,  /3  into  7,  7  into  a.  The  legitimacy  of  this  process  and  the 
truth  of  the  resulting  formulas  appear  from  the  consideration  that 
no  distinction  is  made  as  to  any  one  side  or  any  one  angle.  Any 
side  and  its  opposite  angle  can  be  exchanged  for  any  other  pair. 
The  cyclic  advance  affords  a  convenient  systematic  method  of 
writing  all  possible  forms. 

From  any  one  of  these  sets,  as  for  instance  that  of  Art.  40, 
or  that  of  Art.  42,  all  the  other  sets  may  be  derived  by  purely 
analytical  processes.  An  independent  geometric  proof  is  given  of 
each,  however.  The  derivation  by  the  analytic  method  suggested 
will  afford  a  valuable  review  exercise  after  Chapter  VIII  has  been 
studied. 

62 


LAWS   FOR  OBLIQUE   TRIANGLES 


53 


40.  Law  of  projections.  If,  in 
Fig.  30,  the  perpendicular  CD  is 
drawn  from  0  to  AB^  the  portions 
AB  and  BB  are  respectively  the 
projections  on  the  side  AB  of  the 
other  two  sides  AC  and  CB.  Con- 
sequently, by  Art.  15,  we  have 


Fig.  30. 


AB  ==ACco^a-\-CB  cos  /3, 

or  c  =  &  cos  a  +  a  cos  p. 

By  drawing  the  perpendicular  from  A  and  B  in  turn,  we  get 

a  zz  c  cos  p  +  6  cos  "y, 

6  =  a  cos  "Y  +  c  cos  a. 

By  cyclic  advance  of  the  letters  the  first  formula  is  transformed 
into  the  second,  the  second  into  the  third,  and  the  third  into 
the  first. 

41.    Law  of  sines.      Connect  the  circumcenter  K  in  Fig.  31 
with  the  vertices,  A^  B,  C,  and  the  midpoints,  X,  M,  iV,  of  the  sides. 

Then  is  Z  BK0  =  2  «,  Z  CKA  =  2  /3, 
Z.AKB=2y.  (Why?)  In  the 
right  triangle  KLO^  /.  LKO  =  a, 
and  LC=^a.  Denoting  the  cir- 
cumradius  by  R^  Art.  16  gives 

R  sin  a. 


Fig.  31. 


J- a 


The    other    right     triangles     give 
likewise 

^b  =  R  sin  yS, 

^  c  =  R  sin  y. 


Equating  the  values  of  2  R,  we  obtain  the  law  of  sines ;  namely, 

a    _    b    _     c 
sin  a     sin  p     sin  'y 

The  cyclic  symmetry  is  apparent. 

The  student  should  draw  the  figure  and  give  the  proof  in  case 
one  angle  of  the  triangle  is  obtuse. 

42.    Law  of  cosines.     In  Fig.  32  the  perpendicular  jt?  drawn  from 
(7  divides  the  opposite  side  c  into  two  portions  m  and  w,  and  the 


54 


OBLIQUE   TRIANGLES 


whole   triangle   into  two   right    triangles  ADC  and  BDO,     In 

the    latter    triangles,    we   have,   by 
Art.  16, 

a^  =  n^  -\-p^ 

=  ((?  —  771)2  -{-p^ 

Fig.  32.  or     a^  —  ^2  ^  ^2  _  2  hc  COS  a. 

Proper  changes  in  the  figure  yield 

62  =  c'^  +  a^  -2ac  cos  p, 
c2  =  a2  _f.  ^2  -2  ah  cos  7. 

These  again  may  be  written  by  cyclic  advance  of  the  letters. 
Useful  forms  for  writinsr  these  laws  are : 


cos  a  = 

52+^2- 

a^ 

2  be 

cos  /3  = 

c2-fa2- 

h'^ 

2  ao 

i->r»o   i\/  — 

a'^+P- 

-6'2 

lab 

43.    Law  of  tangents.     lu  Fig.  33  draw  AE  the  bisector  of  the 

angle  at  J.,  and  BF  and  CD  perpendicular  to  it  from  the  other 

vertices. 

Then 

ABAF=^DAC=^a, 

while 

Z  DCE=  Z.  EBF=  90°  -  Z  BEF 

=  90°  -  (Z  ABE  +  Z  ^^  JS') 

=  K«  +  ^  +  7)-(^  +  -|«) 

=  K7-/3). 
Again, 

i)^=  ^^+  DE=AF-  AD. 

From  the  right  triangles  in  the  figure  we  get 

AF~AD 


Fig.  33. 


(e- 


b)  cos  J  a 


^^^2^7     Z:^;      ^^      ^^^      FB  +  CD     FB^-CD      ((?  +  *)  sin  i« 


or 


taiil(v-p) 


c  +  6 


cot  ^  a. 


LAWS   FOR   OBLIQUE   TRIANGLES 


55 


The  forms 


tani  (a-v)  =  ^-^cotlp, 


a 


tani(p-a)=— --cot|Y, 


&  +  « 

may  be  obtained  from  suitably  altered  figures  or  by  cyclic  advance. 
The  formula  may  be  written  symmetrically 

tanK7-y3)^g-^ 
tan i  (7  +  yg)      c-^h 

Ji  b>c,  the  first  formula  will  stand 

tan  i  (/?  -  ry)  =  -Zl  cot  I  a. 

Similar  changes  may  occur  in  the  other  two. 

44.   Angles  in  terms  of   the  sides.     Construct   the   inscribed 
circle,    Fig.    3-4,    and    denote    its 
radius  by  r.    Denoting  the  perim- 
eter a-{-b-{-  c  by  2  s,  we  have 

AE=AF=s-a, 

BD  =  BF=s-h, 

CD=CE  =  s-  c. 

Consequently,  by  Art.  16, 

V  V 

tan  J  a  =  ,  tan  \  p  = -,  tan  \  y  = 


The  value  of  r  in  terms  of  the  three  sides  is  derived  in  the 
corollary  of  Art.  45,  thus  completing  this  theorem. 

45.    Area  of  oblique  triangles. 

(1)  By  elementary  geometry,  we 
have  (see  Fig.  35) 

Introducing  the  value  of  p  found  by 
Art.  16,  we  get  the  formula 

^  =  1 6c  sin  a. 


56  OBLIQUE   TRIANGLES 

with  the  cognate  forms 

A=  ^ca  sin  p,    ^  =  |  ab  sin  -y. 

(2)  Squaring  both  members  of  the  formula  just  derived,  we 
obtain,  with  the  aid  of  readily  justifiable  transformations  and  sub- 
stitutions. 


=  i5V(l-cos2 

a) 

, 

=  — (l  +  coscc)  • 

|(1- 

-  cos  a) 

her.  ^h^  +  e^- 

-> 

!(■ 

b'^^c^- 
2  5c 

.■) 

2bc-hb^  +  c^- 

aP'    2bc- 

52_,.2  4.^2 

4 

4 

b-\-c  +  a     b  + 

c—  a 

a- 

-b-{-  c      a-\-  b  —  c 

2  2 

=  s(s  —  «)(s  —  5)(s  —  c). 

Whence  we  have  the  desired  formula 


A  =  Vs(s  —  a)(s  —  h)  (s  —  c). 
(3)  If  r  is  the  radius  of   the  inscribed  circle,  we  have,  by 
elementary  geometry, 

A  =  rs. 

Corollary.  Equating  the  values  of  A  found  in  (2)  and  (3), 
and  solving  for  r,  we  get 

^  s 

the  result  needed  to  complete  the  theorem  of  Art.  44. 

46.  Numerical  solution.  The  formulas  of  Arts.  40  and  42  are 
not  adapted  to  the  employment  of  logarithms.  They  are  useful, 
however,  in  case  the  numerical  values  of  the  sides  contain  few 
digits. 

The  solution  of  oblique  triangles  falls  into  four  well-defined 
cases,  according  as  the  three  given  parts  consist  of 

I.  Two  angles  and  one  side. 

II.  Two  sides  and  an  angle  opposite  one  of  them. 

III.  Two  sides  and  the  included  angle. 

IV.  Three  sides. 


NUMERICAL  SOLUTION.    CASE  I  57 

Each  of  these  three  cases  with  a  model  solution  is  discussed  in 
detail  in  the  following  articles. 

47.  Case  I.  Given  two  angles  and  one  side.  Let  the  given 
parts  be  a,  j3,  a. 

The  solution  is  effected  by  means  of  the  formulas  of  Arts.  39 
and  41.     Solving  for  the  unknown  parts,  we  have 

7  =  180°-(«  +  /3), 
,      a  sin  y8 


Example. 


sin  a  ' 

a  sin  7 
c=—. -, 

sin  a 

t,          ,                   b  sin  7 
formula          c  =  —. — ~ . 

sm  l3 

Given           a  =  47°  13.2' 

^=65°  24.5' 

a  =  43.176 

sum  of  angles  =  180° 

«  4-/5  =  112°  37.7' 

.-.  7=    67°  22.3' 

loga=    1.63524 

log  sin /3=    9.95871- 

-10 

cologsina=    0.13433 
log  5  =  11.72828 - 

-10 

.-.  5  =  53.491 

loga=    1.63524 

log  sin  7=    9.96522- 

-10 

cologsina=    0.13433 
log  c=  11.73479- 

n^ 

.-.  c  =  54.299 

Check 

log  5=    1.72828 

log  sin  7=    9.96522- 

10 

cologsiny8=    0.04129 
log  (?  =  11.73479- 

-10 

/.  (?=  54.299 

58  ^  OBLIQUE   TRIANGLES 

The  compact  form  of  computation  is  as  follows ; 

log  1.63524 


a  =  43.176 
^  ^  65°  24.5' 
a  =  47°  13.2' 
b  =  53.491 
y  =  67°  22.3 
c  =  54.299 


log  1.63524 
log  sin  9.95871  -  10 
colog  sin  0.13433 


log  b  1.72828 


coiog  sin  0.13433 
log  sin  9.9652g  -  10 


Check 
colog  sin  0.04129 

log  1.72828 
log  sin  9.96522 


log  c  1.73479 


Examples 
Find  the  remaining  three  parts,  given 

1.  ;8  =  65°15.5',        y  =  81°  24.6', 

2.  /?  =  38°37.4',        7  =  75°  32.8', 

3.  a=  48°  29.2',        y=  115°  33.8', 
^^—     4.    a  =  68°  41.5',        y  =  110°  16.5', 


10 


log  c  1.73479 


b  =  724.32. 
c  =  129.63. 
a  =  14.829. 
c  =  9.4326. 


48.  Case  II.  Given  two  sides  and  an  angle  opposite  one. 
Let  the  given  parts  be  a,  5,  a. 

The  solution  is  effected  by  the  formulas  of  Arts.  39  and  41. 

Solving,  we  have 

.     ^      b  sin  a 
sin/3=  — , 

a  sin  7 


c  = 


with  the  check  formula 


(?  = 


sm  a 

5  sin  7 
sin  y8 


An  ambiguity  arises  in  this  case,  however,  since  to  any  value 
of  the  sine  correspond  two  supplementary  angles,  one  acute,  the 
other  obtuse.     Thus  we  also  have 


/3' 

=  180°-/3, 

i 

=  180°-  (a 

+  /30. 

c' 

a  sin  7' 
sin  a   ' 

e' 

h  sin  7' 

siuiS' 


CASE  II 


59 


The  nature  of  this  ambiguity  will  appear  from  the  construction 
of  the  triangles  with  the  given  parts.  If  the  given  angle  a  is 
acute,  there  will  be  no  solution,  one  solution,  or  two  solutions, 
according  as  the  free  end  of  a  (see  Fig.  36),  swinging  about 


.>— . 


A\ 


T^L 


Fig.  36. 

(7,  meets  the  line  AL  in  no  points,  one  point,  or  two  points ; 
i.e.  as  a  is  shorter  than  (72),  the  perpendicular  from  0  upon  AL, 
longer  than  AO^  or  intermediate  between  CD  and  AC  For 
a  =  CD  there  is  a  single  right  triangle ;  and  for  a  =  AC,  a  single 
isosceles  triangle. 

When  a  is  right  or  obtuse,  there  is  no  solution  or  one  solution, 
according  as  a  is  shorter  or  longer  than  AC. 

These  results  may  be  tabulated  for  reference. 


«<90' 


'a<h  sin  a, 
b  sin  a<a<h^ 

a  =  b  sin  a,j 


no  solution, 
two  solutions, 

one  solution. 


> 


90^ 


a  :^  6,     no  solution, 
>  5,     one  solution. 


If  we  proceed  with  the  numerical  work,  without  previously 
testing  the  number  of  solutions  possible,  the  case  of  a  single 
solution  will  appear  from  the  fact  that  a -\-  jS^  >  180°.  (Whence 
a  +  (180°  -  /9)  >  180°,  or  a  -  yS  >  0,  or  ^  <  a.)  When  there  is  no 
solution,  we  shall  get  log  sin  fi>0 ;  i.e.  its  augmented  character- 
istic will  be  10  or  greater.  A  preliminary  free-hand  sketch  will 
ordinarily  serve  to  determine  the  number  of  possible  solutions. 


Example  i.     Given 


a  =  3541, 
b  =  4017, 
a  =  61°  27'. 


60 


OBLIQUE  TRIANGLES 


By  careful  arrangement  of  the  work,  we  can  determine  the 
number  of  solutions  by  inspection. 

Check 
log 


&=4017 
az=61°27' 

h  sin  a 

a  =  3541 
^=  85°  ir 

a-f;8=146°38' 
7  =  33°  22' 
c=2217.16 
c  =  2217.16 


3.60390 
log  sin  9.94369-10 
log        3.54759 
log        3.54913 
log  sin  9.99846 


colog  sin  0.05631 
log         3.54913 


log  sin  9.74036-10 
log        3.34580 


log 


3.60390 


colog  sin  0.00154 
log  sin  9.74036-10 
log        3.34580 


From  the  logarithms  of  6,  a,  and  h  sin  a  it  is  seen  that  h  sin  a 
<a<h,  whence  there  are  two  solutions.  For  the  second  solution 
we  have  : 


a=  61°  27' 
)8'=  94°  49' 
«  +  /?'  =  156°  16' 
y'  =  23°44' 
a  =  3541 
6  =  4017 
c'  =  1622.52 
c'  =  1622.52 


colog  sin  0.05631 


log  sin  9.60474  - 10 
log        3.54913 


log 


3.21018 


Check 
colog  sin  0.00154 
log  sin     9.60474  - 10 
log  3.60390 

log  3.21018 


Example  2.  How  many  triangles  are  determined  by  the 
given  parts  a  =  30°,  h  =  24,  «  =  10,  12,  20,  24,  30  ? 

Here  6  sin  a  =  24  x  J  =  12.  Accordingly,  we  have,  for  a  =  10, 
no  triangle  ;  for  «  =  12,  one  right  triangle  ;  for  a  =  20,  two  triangles  ; 
for  a  =  24,  one  isosceles  triangle  ;  and  for  a  =  30,  one  triangle. 

Examples 

1.  How  many  triangles  are  determined  by  the  given  parts  (3  =  43°,  c  =  120, 
and  h  =  63,  81.884,  95,  120,  150? 

2.  How  many  triangles  are  determined  by  the  given  parts  y  =  54°,  a  =  75, 
and  c  =  51,  60,  67.5,  70,  75,  100? 

Find  the  remaining  parts  of  all  possible  triangles,  given 


3.  a 


62.518, 


4.  a=  429.15, 

5.  6  =  3912.7, 

6.  6  =  129680, 


6  =  72.932, 
c=    328.12, 
c  =  3526.5, 
c  =  152960, 


/3=  98°  23.5'. 
a  =  130°  33.7'. 
y=  35°  25.8'. 
13=   38°  28.8'. 


CASE   III 


61 


49.    Case  III.     Given  two  sides  and  the  included  angle.     Let 

the  given  parts  be  «,  6,  7,  with  a>h.     The  solution  is  effected  by 
the  formulas  of  Arts.  43,  39,  and  41.     Solving,  we  have 


tan  1  («  —  /3)  = 


I(«  +  ^)  =  90°-i7, 
a  sin  7 


cot  J  7, 


with  the  check  formula 
Example.     Given 


y  =  78°  15' 

a  =  .745 

6  =  .231 

a-/;  =.514 

log        9.71096- 

-10 

a +  6=. 976 

colog     0.01055 

^=39°  7.5' 
2 

log  cot  0.08969 

^^^=32°  55.3' 
2 

^^±^=50°  52.5' 
2 

log  tan  9.81120 - 

-10 

a  =  83°  47.8' 

)8  =  17°57.2' 

c  =  . 73368 

c  =  . 73367 

sm  a 

h  sin  7 
smy3 

a  =.745, 
5  =  .231, 
7  =  78°15^ 


log  sin  9.99080- 10 
log        9.87216-10 


colog  sin  0.00255 
log         19.86551-20 


Examples 

Find  the  unknown  parts,  given 

1.  6  =  284.12,        c  =  361.26,  a  =  125°  32'. 

2.  c  =  395.71,         a  =  482.33,  ^  =  137°  21'. 

3.  a  =  .06351,         c  =  .10329,  /8  =  83°  29.4.' 

4.  c  =  .00397,         h  =  .00513,  a  =  68°  21.8^ 


log  sin     9.99080-10 
log  9.36361  - 10 


colog  sin  0.51109 


19.86550-20 


50.    Case  IV.     Given  the  three  sides.     The  given  parts  are 
a,  5,  G. 


62 


OBLIQUE   TRIANGLES 


The  solution  is  effected  by  the  formulas  of  Art.  44,  with  the 
formula  for  r  from  Art.  45.      We  have  at  once 

s  =  1  (a  -f  6  +  0' 


^_J(«-«)(«-*)(« 

-0 

tan  -«  = ,  etc. 

2        s  — a 

«  +  /3  +  7  =  180°,  serves  as  a  check  formula. 

Example  i.     Given 

a  =.05341, 

5  =.06217, 

tf=.  03482. 

Then                   2  s  =  .15040 

s=  .07520 

colog 

1.12378 

s-a=. 02179 

log 

8.33826-10 

s- 5  =  .01303 

log 

8.11494-10 

«_  ^  =  .04038 

log 

8.60617-10 

r2 

log 

16.18315-20 

r 

log 

8.09157-10 

^=29°  32.3' 

log  tan 

9.75331-10 

1  =  43°  27.6' 

log  tan 

9.97663-10 

^=17°  0.1' 

log  tan 

9.48540-10 

a=    59°    4.6' 

•       y8=    86°  55.2' 

7=    34°    0.2' 

sum  of  angles  =180°       0' 

When  the  three  sides  are  given  and  only  one  angle  is  required, 
say  /S,  the  two  appropriate  formulas  may  be  combined  into  one,  as 


tan  -^ 


_^l(8  —  a)(8  —  c) 


sCs-h) 


CASE  IV.     COMPOSITION  OF  FORCES  63 

Example  2.     Given 

a=    35, 

h=    64, 

c=    73. 

Then  2  s  =  172 

s=    86  colog     8.06550-10 

s-a=    51  log     1.70757 

8-h=    22  colog     8.65758-10 

8-e=    13  log     1.11394 

2)19.54459-20 
i;8=30°37.4'    log  tan     9.7723U-10 

;e=61°14.8' 

Examples 
Find  the  angles  of  the  following  triangles : 

1.  a  =  6123,       ^>  =  7148,       c  =  6815. 

2.  a  =  12,545,     5=8612,        ^=10,353. 

3.  a  =  .05431,    5  =  .03714,    6'=. 06513. 
-- — 4.    a  =  .006152,  b  =  .008174,  c  =  .007534. 


5.    ^.  =  72,584, 

5  =  125,217,  c?=  36,925. 

6.    a  =  13,579, 

6  =  35,791,     (?  =  24,680;  find /3. 

7.    a  =  80,812, 

h  =  37,194,    e  =  43,618. 

8.    «  =  36,925, 

5  =  25,814,     c=  14,703;  find  7. 

Find  the  areas  in  examples  1  and  2. 

51.    Composition   and   resolution   of  forces.     Equilibrium.     In 

mechanics  the  solution  of  oblique  triangles  is  frequently  required 
in  problems  relating  to  the  composition  and  resolution  of  forces, 
velocities,  and  other  directed  quantities. 

In  this  article  will  be  stated,  without  proof,  some  of  the  laws 
governing  the  combination  of  such  quantities,  showing  the  appli- 
cation of  trigonometry  to  certain  of  the  problems  involved. 

Suppose  the  line  segments  AB  and  JL(7,  P'ig.  37,  to  represent 
in  magnitude  and  direction  two  forces  acting  at  a  point  J.,  and  in- 
cluding between  their  lines  of  action  the  angle  <^. 


64  OBLIQUE   TRIANGLES 

Complete  the  parallelogram  ABBQ.  The  diagonal  AB^  drawn 
from  the  point  A^  is  the  line  segment  representing  the  resultant 

of  the  two  given  forces,  i.e.  the  sin- 
gle force  that  will  produce  the  same 
effect  as  the  two  given  forces.  The 
process  of  finding  the  resultant  of 
two  or  more  given  forces  is  called 
the  composition  of  forces. 

Conversely,  the  two  line  segments 
AB  and  AC  may  be  taken  as  the 
components  of  AB.     Thus  the  two 
Fig.  37.  ^    forces  AB  and  AC,  acting  together 

at  A,  produce  the  same  effect  as  the  single  force  AB.  The  pro- 
cess of  finding  two  or  more  forces  equivalent  to  a  given  force  is 
called  the  resolution  of  the  force  into  its  components. 

Since  the  segment  BB  is  equal  and  parallel  to  AC,  it  follows 
that  the  resultant  and  the  two  components  form  a  closed  triangle 
ABB,  and  the  relation  between  the  forces  may  be  obtained  by 
solving  this  triangle.  Note  that  the  angle  ABB  is  the  supple- 
ment of  the  angle  <^,  so  that  by  Art  37, 

cos  ABB  =  —  cos  (/). 
Example  1.    Find  the  resultant  of  two  forces  of  320  dynes 
and  400  dynes,  respectively,  acting  on  a  common  point,  at  an  angle 
of  54°  28^ 

In  the  triangle  ABB,  Fig.  37,  we  have  given  two  sides  and 
the  included  angle.  If  only  the  magnitude  of  the  resultant  is 
desired,  it  may  be  obtained  by  the  law  of  cosines.  Art.  42.     Thus 

we  obtain  „       « 

AB  =  ^\A^  -f  ^(7+2  AB    AC  ■  cos  c^j. 

If  the  angle  formed  by  the  resultant  with  its  components  is  also 
required,  the  logarithmic  computation  may  be  effected  as  in  Case 
III,  Art.  49. 

Example  2.  Resolve  a  force  of  40  pounds  into  components 
making  angles  of  32°  and  74°  20  with  its  line  of  action. 

Referring  to  Fig.  37,  we  have 

^D  =  40,  Z  ^^2>  =  32°,  and  Z  i>^  (7=  Z  ^i)^  =  74°  20' . 

Denoting  the  sides  opposite  the  angles  A,  B,  B,  respectively,  by 

a,  h,  d,  we  have  from  the  law  of  sines, 

,sin^  J      ysini) 

a  —  b -,  d  =  o— — — • 

sin  B  sin  ^ 


EQUILIBRIUM  OF  FORCES 


65 


Hence  the  components  may  be  computed. 

Three  forces  are  in  equilibrium  when  the  resultant  of  any  two 
forces  is  equal  and  opposite  to  the  third.  Thus  in  Fig.  37,  if 
the  direction  of  the  force  AD  is  reversed,  it  and  the  forces  AB 
and  AQ  will  be  in  equilibrium.  The  necessary  conditions  that 
three  forces  shall  be  in  equilibrium  are  : 

1.  Their  lines  of  action  shall  lie  in  the  same  plane. 

2.  Their  lines  of  action  shall  meet  in  a  point. 

3.  The  line  segments  representing  the  three  forces  when  laid 
off  in  order  shall  form  a  triangle. 

In  Fig.  38  the  forces  a,  5,  and  c  applied  at  a  common  point  are 
in  equilibrium.  The  angles  between  the  lines  of  action  are  de- 
noted by  J.,  B^  C^  as  indicated.     When  the  forces  are  laid  off  to 


form  the  triangle,  the  angles  of  the  triangle  are  seen  to  be  the 
supplements  of  the  corresponding  angles  A,  B^  O. 

That  is, 

a  =  180°  —  J.,  whence  sin  a  =  sin  A, 

/3  =  180°  -  B,  whence  sin  /S  =  sin  B. 
etc.  etc. 

From  the  law  of  sines. 


Therefore, 


a            h 

c 

sin  a      sin  ^ 

sin  7 

a              h 

c 

sin  A      sin  B      sin  O 


66 


OBLIQUE   TRIANGLES 


EXERCISE    XIV 
Find  the  unknown  parts  of  the  following  triangles : 


a 

18 

y 

a 

b 

c 

1. 

62°  35' 

82916 

59278 

2. 

75290 

92841 

69289 

3. 

25°  36.2' 

68°  13.5' 

3.9168 

4. 

55°  55.4' 

.25317 

.36291 

5. 

69°  17.5' 

329.12 

689.12 

6. 

100°  10' 

62198 

29322 

7. 

.0000713 

.0000987 

.0001255 

8. 

61°  15.2' 

49°  16.3' 

58.291 

9. 

120°  50.2' 

2.8315 

4.1217 

10. 

38°  17.2' 

21.992 

50.715 

11. 

150°  24.2' 

.038251 

.047319 

12. 

58°  06.5' 

57.15 

67.31 

13. 

75°  19.3' 

70°  29.2' 

658.42 

14. 

100.05 

200.07 

150.08 

15. 

126°  26.4' 

.0021868 

.0032292 

16. 

10°  32.8' 

25.317 

37.293 

17. 

50010 

70020 

90030 

18. 

48°  25.3' 

56°  34.5' 

7219.2 

19. 

120°  15' 

62158 

75292 

20. 

90°  00' 

725.63 

617.25 

Solve  the  following  triangles,  given 
21.  a  =  2500,  c  =  2125,  A  =  208,690. 
22..  ft  =  103.5,       c  =  90,         A  =  4586.7. 

23.  a  =  73°  10',     b  =  753,       A  =  74,803. 

24.  ^  =  57°  25',     c  =  57.65,     A  =  3055.7. 

25.  Find  the  areas  in  examples  1,  9,  17. 

26.  Find  the  areas  in  examples  2,  4,  14. 

27.  Determine  the  magnitude  and  direction  of  the  resultant  of  two  forces 
of  magnitudes  a  and  h,  if  their  lines  of  action  include  an  angle  <f>. 

28.  Carry  out  the  computation  of  example  27  in  the  following  cases : 

a  =  20,  &  =  36,  <^  =    45° ;    a  =  300,  b  =  540,  <^  =    64°; 
a  =  75,  6  =3  60,  <^  =  145° ;    a  =  250,  b  =  320,  <^  =  120°. 

29.  Find  the  directions  of   three  forces  in  equilibrium  if   a  =  7,  6  =  10, 


c  =  15;  also  if  a  =  24,  6  ==  36,  c  =  42. 


EXERCISES 


67 


30.   Referring  to  Figure  i 
a  =  695,  6  =  483,  0  =  155°:  a 


'  solve  completely  and  interpret  physically  when 
720,  b  =  840,  B  =  100°. 

=  135, 


31.  Solve   and   interpret    when   a  =  1200,    5  =  135°,    C  =  150° 
h  =  142,  c  =  95. 

32.  Resolve  a  force  of  magnitude  84  into  two  equal  components  making  an 
angle  of  60°  with  each  other. 

33.  Resolve  a  force  of  magnitude  240  into  two  components  of  120  and  180 
each  and  find  the  directions  of  the  components. 

34.  Determine  the  formula  for  one  side  of  a  quadrilateral  in  terms  of  the 
other  three  sides  and  their  included  angles.  Compute  for  a  =  10,b  =  12,  c  =  15, 
^6  =  135°,  6c  =  60°. 

Query.  How  many  given  parts  serve  to  determine  the  remaining  parts 
of  a  quadrilateral? 

35.  Given  the  four  sides  and  one  angle  of  a  quadrilateral,  determine  the 
other  angles  and  the  diagonals.  Compute  for  a  =  QO,  b  =  72,  c  =  90,  d  =  100, 
Q  =  120°. 

36.  Given  three  angles  and  two  sides  of  a  quadrilateral,  determine  the 
remaining  sides.     Compute  for  a  =  630,  b  =  500,  ab  =  100°,  be  =  80°,  cd=  60°. 

37.  Find  the  angles  and  the  lengths  of  the  sides  of  a  regular  pentagram, 
or  five-pointed  star,  inscribed  in  a  circle   of   radius  8. 

38.  Compute  the  volume  for  each  foot  in  depth  of  a 
horizontal  cylindrical  tank  of  length  30  feet  and  radius 
6  feet. 

39.  Having  measured  the 
following  data,  ^A  =  80°  30', 
B  =  72°  15',  and  c  =  232.5 
feet,  compute  the  inaccessi- 
ble distance  b  (Fig.  39). 

40.    Compute  the  dis- 
tance a  across  a  lake.  Fig. 
40,    having    measured    A, 
B,  and  c,  which  are  respectively  51°  20',  72°  40'  and    , 
3420.5  feet. 


Fig.  41. 


41.  A  being  invisible 
from  C,  find  the  distance  b 
through  a  forest,  having 
measured   a  =  1037  feet,  c  =  1208   feet,   B  =  69°  25'. 

42.  In  Fig.  42,  BC,  the  distance  of  the  foot  of 
a  wall  below  the  instrument  is  12.3  feet,  6  and  a, 
the  angles  of    elevation  and  depression,   are  15°  20' 

and  21°   15',   respectively.      Find  the  height  of   the   wall    and   its   distance 
from  the  instrument. 


Fig.  42. 


68 


OBLIQUE   TRIANGLES 


Fig.  43. 


Fig.  44. 


43.  A  pole  BC,  Fig.  43,  is  12  feet  long  and  leans  two  feet  from  a  vertical 
toward  the  instrument  at  ^ .  If  the  angles  of  elevation  of  the  top  and  bottom 
are  respectively  37°  15'  and  11°  50',  what  are  the  horizontal  and 
vertical  distances  from  the  instrument  to  the  foot  of  the  pole? 

44.    It  is  desired  to  find  the 

horizontal    distance    and   eleva- 
tion  of    the    inaccessible 

point   B,   Fig.  44, 

with    reference    to 

an    instrument    at 

A.      Having    laid 

out     a    base     line 

A  C,  250  feet  long, 

the  angles  at  A  and  C   are  found  to  be  87°  10' 
and  73°  51',  respectively,  and  from  A  the  angular  elevation  of  B  is  11°  32'. 

,(7  45.   Given    5  =  110°  05',    ^  J5:  =  .4 7)  =  200  f eet, 

DE  =  125  feet,  and  ^5  =  632  feet;  find  the  distance 
AC  to  be  laid  off,  and  the 
inaccessible    distance    BC 
(Fig.  45). 

46.  From  measure- 
ments we  have  (Fig.  46) 
AB  =  Qm  feet,  BAC  =  10°  40',  BAD  =  Q2°  30', 
ABD  =  65°  32',  ABC  =  89°  25'.  Find  the  inacces- 
sible distances  AD  F^«-  ^^^ 
and  DC,  and  the  angle  between  DC  and  AB. 

47.  From  the  instrument  at  A  (Fig.  47) 
the  angles  of  elevation  to  the  top  and  base  of 
the  vertical  wall  are  15°  12'  and  1°  23',  respec- 
tively. A  base  line  AB  \&  measured  75  feet 
toward  the  wall  down  a  plane  inclined  8°  16', 
and  from  B  the  angle  of  elevation  to  the  top 
of  the  wall  is  37°  46'.  Compute  the  height  of 
the  wall  and  its  horizontal  distance  from  A. 


Fig.  47. 


48.  It  is  required  to  prolong  the  line  AB  (Fig.  48)  beyond  an  obstacle. 
At  B  is  made  an  angle  52°  20'  to  the  right 

and  at  C  an  angle  of  110°  00'  to  the  left,  BC 
being  210  feet.  Compute  the  proper  distance 
CD  and  angle  to  the  right  at  Z),  also  the 
inaccessible  distance  BD.  Note  that  by  mak- 
ing B  =  D  =  60°  and  C  =  120°,  then  BC=CD 
=  BD  and  all  computations  are  avoided. 

49.  Having  but  one  point  C  (Fig.  49)  from  which  both  inaccessible  points 
A  and  B  are  visible,  we  are  required  to  find  the  inaccessible  distances  AC 


EXERCISES 


69 


and  AD  and  the  angle  between  AB  and  DC. 
ADC  =  87°  42',  DC  A  =  60°  32',  DCE  =  170°  05', 
BCE  =  41°  20',  CEB  =  111°  35',  DC  =  365.2  feet, 
C^^  =  410.7  feet. 

50.    It  is  required  to  ascertain  the  length  and   j) 

position  of  an  in- 
A_, ,B 


Fig.  50. 


accessible  line  AB 
(Fig.  50),  its  ex- 
tremities not  being  visible  from  a  common 
point  beyond  the  obstacles.  By  chaining 
we  have  CD  =  210.7  feet,  DE  =  390.4  feet, 
EF  =  173.5   feet. 


Then  the  follow- 
ing angles  are  measured :  A  CD  =  83°  41',  CDE  = 
19°  12'  left  (180°-19°  12'),  CDA  =  79°  49',  FEB  = 
53°  20',  DEF  =  42°  03'  left,  EFB  =  115°  27'. 

In  order  to  locate  points  suitably  upon  a  map, 
find  lengths  AB,  AD,  and  BE. 


/77777my 


51.   A  tower  115  feet  high  casts  a  shadow  157 
feet   long  upon  a  walk  which   slopes   downward  Fig-  51. 

from  its  base  at  the  rate  of  1  in  10.     What  is  the  elevation  of  the  sun  above 
the  horizon? 


CHAPTER   YII 

THE  GENERAL  ANGLE 

Only  those  parts  of  trigonometry  that  are  necessary  for  the  solution  of  triangles 
have  been  developed  thus  far.  In  this  and  the  following  chapters  are  considered 
some  of  the  more  important  topics  of  another  phase  of  trigonometry  that  is  no  less 
essential  for  the  further  study  of  pure  and  applied  mathematics. 

52.  General  definition  of  an  angle.  If  a  straight  line  rotates 
about  one  of  its  points,  remaining  always  in  the  same  plane,  it 
generates  an  angle.  The  angle  is  measured  by  the  amount  of  ro- 
tation by  which  the  line  is  brought  from  its  original  position  into 
its  terminal  position.  For  the  small  rotation  leading  to  acute  and 
obtuse  angles  this  definition  agrees  with  the  customary  elementary 
definition,  the  knowledge  of  which  has  been  presupposed  in  the 
foregoing  chapters. 

As  in  Art.  3,  counterclockwise  rotation  generates  positive 
angles ;   clockwise  rotation,  negative. 

In  the  sexagesimal  system  of  angle  measurement  the  standard 
unit  is  the  angle  produced  by  one  complete  rotation  of  the 
generating  line.  This  angle  is  divided  into  360  equal  parts 
called  degrees^  the  degree  into  60  minutes,  and  the  minute  into 
60  seconds. 

In  the  circular  system  the  standard  unit  is  the  radian^  the 
angle  produced  by  such  a  rotation  that  each  point  in  the  generat- 
ing line  describes  an  arc  equal  in  length  to  its  radius.  Angu- 
lar magnitudes  are  stated  in  radians  and  decimal  fractions 
thereof. 

Instruments  are  graduated  and  tables  printed  in  accordance 
with  the  sexagesimal  system,  which  is  used  in  practical  numerical 
calculations.  Astronomers,  however,  employ  decimal  fractions  of 
seconds,  while  engineers  make  use  of  tenths  of  minutes  and  deci- 
mal divisions  of  degrees.  In  theoretical  discussions  the  radian 
system  is  commonly  employed.  Hereafter,  in  this  book,  the  two 
systems  will  be  used  interchangeably. 

70 


DEFINITIONS   OF   TRIGONOMETRIC   FUNCTIONS  71 

.^^-^ 

Since  the  circumference  of  a  circle  is  equal  to  2  tt  times  its 
radius,  where  7r=  3.14159---,  we  may  write  the  following  relations 
between  the  two  systems  : 

2  IT  radians  =  360° 

1  radian  =  57.29578°. . 
=  57°  17' 44.8'' 

and,  in  general,  the  number  of  degrees  in  any  angle  is  equal  to 

180 
the  number  of  radians  multiplied  by  • ,  while  the  number  of 

TT 

radians  is  equal  to  the   number   of  degrees  multiplied  by  -^. 

Thus  the  straight  angle  is  tt  radians ;  the  right  angle,  —  radians. 

A 
If  the  radius  of  the  circle  is  represented  by  r,  the  arc  by  a, 
and  the  angle,  in  radians,  by  a,  we  have  the  important  relation 

a  —  vQ^. 

53.  Axes,  quadrants,  etc.  Let  the  two  axes  of  coordinates  be 
assumed  as  in  Art.  4 ;  and,  as  in  Art.  6,  let  the  angle  be  placed 
upon  the  axis,  its  vertex  at  the  origin,  and  its  initial  line 
extending  along  the  X-axis  toward  the  right.  The  sign  and 
magnitude  of  the  angle  will  determine  the  position  of  the  terminal 
line,  causing  it  to  coincide  with  one  of  the  axes  or  to  fall  in  one 
of  the  quadrants.  An  angle  is  said  to  be  of  the  first,  second, 
third,  or  fourth  quadrant  according  as  its  terminal  line  falls  in 
that  quadrant. 

While  the  acute  angle  is  of  the  first  quadrant,  the  converse 
is  by  no  means  necessarily  true.  The  ter- 
minal line  of  every  angle,  however  large, 
must  coincide  with  the  terminal  line  of 
some  positive  angle  less  than  360°  (see  Fig. 
52).  For  the  purpose  of  trigonometry  as 
developed  in  the  present  chapter,  for  every 
angle,  positive  or  negative,  and  of  any  mag- 
nitude, may  be  substituted  a  positive  angle 
less  than   360°. 

54.  Definitions  of  the  trigonometric  functions.  The  trigono- 
metric functions  of  angles  of  any  size  are  defined  identically  as  in 


72 


THE   GENERAL   ANGLE 


Art.  6.     Thus  for  all  positions  of  the  terminal  line,  Fig.  53, 


y       .  X 

-  =  sm  a,         -  =  cos  a, 

V  V 


y  X 

—  =  tan  a,        —  =  cot  a, 

X 


y 


V  V 

-  =  sec  a,        -  =  CSC  a. 
X  y 


W 


O       X 


M 


p\a 


{a) 


Q>)  (c) 

Fig.  53. 


(d) 


55.  Signs  and  limitations  in  value.  The  abscissas  are  positive 
for  all  points  in  the  first  and  fourth  quadrants,  negative  for  those 
in  the  second  and  third.  Ordinates  are  positive  for  all  points  in 
the  first  and  second  quadrants,  negative  for  those  in  the  third  and 
fourth.  The  radius  vector  is,  by  agreement,  considered  positive 
for  all  points. 

In  conformity  with  the  sign  law  of  algebra,  the  functions  of 
angles  of  the  different  quadrants  will  have  signs  as  displayed  in 
the  following  table : 


Quad. 

Sine 

Cosine 

Tangent 

Cotangent 

Secant 

Cosecant 

I 

+ 

+ 

+ 

+ 

+ 

+ 

II 

+ 

- 

— 

— 

- 

+ 

III 

- 

— 

+ 

+ 

— 

— 

IV 

- 

+ 

— 

— 

+ 

- 

It  will  be  noticed  that  for  angles  of  the  first  quadrant  all  six 
functions  are  positive.  In  each  of  the  other  quadrants  one  pair  of 
mutually  reciprocal  functions  are  positive,  the  other  two  pairs  are 
negative.  These  positive  pairs  run  as  follows  :  second  quadrant, 
sine  and  cosecant :  third  quadrant,  tangent  and  cotangent :  fourth 
quadrant,  cosine  and  secant. 


SIGNS  AND  LIMITATIONS  IN  VALUE  73 

The  student  should  establish  these  statements  regarding  the 
signs  of  the  functions  and  memorize  them. 

Since  the  lengths  of  the  abscissa  and  ordinate  can  never  exceed 
that  of  the  radius  vector,  it  follows  that  the  sine  and  cosine  can 
never  be  numerically  greater  than  unity,  and  the  secant  and 
cosecant  can  never  be  numerically  less  than  unity.  The  tangent 
and  cotangent  can  have  numerical  values  either  greater  or  less 
than  unity. 

EXERCISE  XV 

1.  Express  in  degrees,  minutes,  and  seconds  the  angles  — ,  — ,  '-~^, 
o,r«    o«    3^  ■  4       3         6 

2.  Express  in  radians  the  angles  30°,  15°,  45°,  120°,  240°,  300°,  450°. 

/-^r'  In  a  circle  of  radius  60  cm.,  what  is  the  length  of  the  arc  which  sub- 
tends at  the  center  the  angle  30°,  60°,  ^,    ^  ? 
^  '       '    3  '    4 

4.  In  a  circle  of  radius  10  inches,  what  is  the  circular  measure  of  the  angle 
subtended  by  an  arc  whose  length  is  10,  5,  20,  5  ir  inches? 

5.  A  friction  gear  consists  of  two  tangent  wheels,  whose  radii  are  8  and 
12  inches,  respectively.  The  smaller  wheel  makes  4  revolutions  per  second.  Find 
the  number  of  revolutions  per  second  made  by  the  larger,  the  angular  velocity 
of  each,  and  the  linear  velocity  of  a  point  on  the  circumference  of  each.  If 
the  larger  wheel  is  attached  to  the  rear  axle  of  an  automobile  whose  rear  wheel 
has  a  diameter  of  30  inches,  find  the  speed  of  progress  of  the  machine. 

6.  The  diameters  of  the  front  and  rear  sprocket  wheels  of  a  bicycle  are 
10  inches  and  4  inches,  respectively,  and  the  diameter  of  the  rear  wheel  is  28 
inches.  Find  the  rate  of  pedaling  when  the  bicycle  is  traveling  12  miles  per 
hour,  the  corresponding  angular  velocities  of  the  two  sprocket  wheels,  and  the 
linear  velocity  of  the  chain. 

7.  Determine  the  quadrant  to  which  each  of  the  following  angles  belongs : 

210°,  465°,  745°,  -  830°,  ^,    i^,    -^. 
3    '       4    '  3 

8.  Determine  the  signs  of  the  functions  of  the  following  angles:  240°, 

330°,  400°,  ^,    -'Le,    6^. 
3  4' 

9.  Show  that  the  quadrant  to  which  an  angle  belongs  is  determined  if  the 
signs  of  any  two  non-reciprocal  functions  are  given. 

10.  To  what  quadrant  does  an  angle  belong  if  its  sine  and  tangent  are 
negative ;  its  secant  and  cotangent  positive ;  sine  and  secant  negative ;  tangent 
and  cosine  positive  ? 


74 


THE   GENERAL   ANGLE 


11.  Determine  the  quadrants  of  the  following  angles: 

sin"i|;  arccos  —  j\;  arctanf;  cot-^  —  j\. 

12.  Determine  the  quadrants  of  the  following  angles  : 

sin-i  I  =  cot-i  —  f  ;  arccos  —  j%  =  arccsc  i|. 

13.  For  what  values  of  a  is  sin  a  —  cos  a  positive  ? 

14.  For  what  values  of  cc  is  tan  a  —  cot  a  negative  ? 
15-20.    Find  the  missing  values  in  the  following  table  : 


z 

sin 

cos 

tan 

cot 

sec 

CSC 

QlFAD. 

a 

7 
8 

it 

-II 

--A 

—    15 

II 

III 

III 

IV 

0 

¥ 

IV 

^ 

-¥ 

III 

56.  Variation  of  the  trigonometric  functions.  A  change  in 
the  angle  will  produce  a  corresponding  change  in  the  values  of  the 
coordinates  and  in  their  ratios.  If,  for  convenience,  the  chosen 
point  in  the  terminal  line  of  the  angle  is  maintained  at  a  constant 
distance  from  the  vertex,  the  radius  vector  will  retain  the  constant 
value  +  V. 

As  the  angle  0  increases  continuously  from  0°  to  360°,  the 
abscissa  and  ordinate  vary  continuously  between  the  limits  —  v 
and  4-  v.  As  6  increases  from  0°  to  90°,  x  is  positive  and  decreases 
from  V  to  0 ;  as  6  increases  from  90°  to  180°,  x  is  negative  and 
decreases  (algebraically)  from  0  to  —  v ;  as  ^  increases  from  180° 
to  270°,  X  is  negative  and  increases  from  —v  to  0 ;  and  as  6 
increases  from  270°  to  360°,  x  is  positive  and  increases  from  0  to  v. 
As  6  increases  from  0°  to  90°,  i/  is  positive  and  increases  from  0  to 
y ;  as  ^  increases  from  90°  to  180°,  ^  is  positive  and  decreases  from 
V  to  0  ;  as  ^  increases  from  180°  to  270°,  y  is  negative  and  decreases 
from  0  to  —V  ;  as  0  increases  from  270°  to  360°,  i/  is  negative  and 
increases  from  —  v  to  0.  Upon  introducing  these  varying  values 
into  the  ratio  definitions,  we  are  enabled  to  trace  the  variation  of 
the  trigonometric  functions. 

We  see,  for  example,  that  as  0  increases  from  0°  to  360°, 
tan  6  continually  increases  algebraically,  changing  sign  from 
negative  to  positive  through  the  value  0  as  ^  passes  through  0°, 


GRAPHS   OF   THE   TRIGONOMETRIC   FUNCTIONS 


75 


180°,  and  360°,  and  from  positive  to  negative  by  becoming  infinite 
as  6  passes  through  90°  and  270°.  There  is  an  infinite  discon- 
tinuity in  tan  6>,  for  6  =  90°  and  d  =  270°. 

Query.  Which  of  the  trigonometric  functions  other  than  the  tangent 
become  infinite  and  therefore  discontinuous  ? 

The  student  should  trace  the  variation  of  each  function  in  detail,  stating 
the  narrative  verbally. 

57,  Graphs  of  the  trigonometric  functions.  The  whole  behavior 
of  each  function  can  be  conveniently  represented  by  means  of  the 
graphical  method  already  introduced  in  Art.  4.     Assume  a  pair 


Fig.  54.     Graph  of  sin  6. 

of  axes  of  coordinates,  as  in  Art.  4,  and  along  the  JT-axis  to  the 
right  lay  off  equal  spaces  corresponding  to  the  number  of  degrees 
in  the  angle  6.  At  each  point  in  the  JT-axis  erect  a  perpendicular 
whose  length  is  proportional  to  the  value  of  the  sine  of  that  angle. 
Each  point  thus  determined  has  the  property  that  its  abscissa 
represents  the  angle  6  and  its  ordinate  the  corresponding  value 
of  sin  6.  Now  having  located  a  sufficient  number  of  points,  draw 
through  them  a  smooth  curve.  It  will  be  seen  that  the  value, 
sign,  and  variation  of  the  sign  at  each  instant  is  fully  exhibited 
by  the  ordinate,  position,  and  inclination  of  the  curve  or  graph. 
The  same  may  be  done  for  each  of  the  functions. 

The  graphs  of  the  different  functions  are  here  presented. 
The  student  should  trace  carefully  the  intimate  and  exact  cor- 
respondence of  the  graphical  and  the  verbal  narratives. 


76 


THE   GENERAL   ANGLE 


^T 


Fig.  55.    Graph  of  cos  d. 


-h. 


O 


Fig.  56.    Graph  of  tan  d. 


GRAPHS 


77 


^Y 


X 


Fig.  57.     Graph  of  cot  d. 


/^Y 


O 


Fig.  58.     Graph  of  sec  6. 


78 


THE   GENERAL  ANGLE 
AY 


_7C_ 
-7t  2 


2 


371 
2 


0      ^ 

2 


X 


Fig.  59.     Graph  of  esc  d. 

58.  Functions  of  270°  and  360°.  By  the  method  of  limits  em- 
ployed in  Art.  12,  we  get  the  following  sets  of  values : 

sin  270°  =  -  1,  cos  270°  =  0, 

tan  270°  =  oo,  cot  270°  =  0,  ■ 

sec  270°  =  00.  CSC  270°  =  -  1. 

sin  360°  =  0,  cos  360°  =  1, 

tan  360°  =  0,  cot  360°=  oo, 

sec  360°  =  1,  CSC  360°  =  oo. 

Here  oo  is  used  as  before  to  denote  the  value  of  a  fraction  whose 
numerator  remains  finite  while  its  denominator  approaches  zero. 
The  sign  +  or  —  is  prefixed  to  the  symbol  oo  according  as  tlie 
variable  becomes  oo  through  a  positive  or  a  negative  sequence  of 
values.     In  the  light  of  this  discussion  the  values  of  the  functions 

oih  y.  —(k  any  integer)  may  be  tabulated,  the  upper  of  the  pair 

of  double  signs  arising   when  the   angle  approaches  the  critical 
value  from  below. 


e 

sin  9 

COS  0 

tan  9 

cot  6 

sec  B 

esc  B 

0 

TO 

+  1 

TO 

Too 

+  1 

Too 

f 

-fl 

±0 

±cc 

±0 

±co 

+  1 

TT 

±0 

-1 

TO 

Too 

- 1 

±GO 

It 

-1 

TO 

±cc 

±0 

Too 

-1 

27r 

±0 

+  1 

TO 

T  00 

+  1 

Too 

FUNDAMENTAL   RELATIONS  79 

EXERCISE    XVI 

n 

1.  Trace  the  variation,  as  d  varies,  (a)  of  sin  2  6\  (b)  of  cot  -• 

n 

2.  Trace  the  variation,  as  ^  varies,   (a)   of  tan  2^;   (h)  of  cos  -  • 

id 

3.  Draw  the  graph  of  cos  2  B. 

4.  Draw  the  graph  of  sin  3  B. 

5.  In  what  points  will  a  horizontal  line  \  unit  above  the  JT-axis  intersect 
the  graph  of  sin  Q  ?     Explain  the  significance  of  the  result. 

6.  In  what  points  will  a  horizontal  line  1  unit  above  the  X-axis  intersect 
the  graph  of  tan  B1     Explain. 

7.  If  the  graphs  of  tan  B  and  cot  B  are  drawn  on  the  same  axes  to  the  same 
scale,  where  will  they  intersect?     What  is  the  significance? 

8.  If  the  graphs  of  sin  B  and  cos  B  are  drawn  on  the  same  axes  to  the  same 
scale,  where  will  they  intersect?    What  is  the  significance  ? 

9.  Construct  the  graph  of  logj^x,  taking  values  of  the  number  x  as  abscis- 
sas and  the  corresponding  logarithms  as  ordinates. 

59.    Fundamental  relations.     Just  as  in  Art.  9  we  find,  by  in- 
spection, 

(1) 

(2) 

(3) 

by  division,  tan  a  =  ^*L^^  (^4) 

(5) 
sm  a 

by  virtue  of  the  Pythagorean  proposition, 

sin^  a  +  cos^  a  =  1,  (6) 

tan^  a  +  1  =  sec^  a,  (T) 

cot^  a  -f-  1  =  csc^  a.  (8) 

The  student  should  prove  that  all  these  formulas  conform,  for 
angles  in  all  quadrants,  to  the  algebraic  law  of  signs. 


CSC  a 

1 
sina^ 

sec  a 

1 
cos  a 

cot  a 

1     . 
tan  a' 

tana 

sin  a 
cos  a 

nnt  CL 

_  cos  a . 

80  THE   GENERAL   ANGLE 

60.    Line  representations  of  the  trigonometric  functions.     As  the 

names  tangent  and  secant  indicate,  the  trigonometric  functions 
were  originally  defined  as  certain  lines  measured  in  terms  of  a 
standard  unit  line.  The  adoption  of  the  abstract  ratios,  as  in 
this  book,  is  of  comparatively  recent  date.  It  is  both  interesting 
and  advantageous  to  know  the  line  representations  and  sliow  that 
they  lead  to  the  same  science  of  trigonometry  as  do  the  ratio  deti- 
nitions. 

The  line  representations  most  frequently  used  involve  the  use 
of  a  unit  circle,  i.e.  a  circle  of  radius  unity.  It  is  evident  that 
we  may  replace  each  of  the  defining  ratios  of  Art.  54  by  an  equal 
ratio  so  chosen  that  its  denominator  is  positive  unity.  The  value 
of  the  ratio  will  be  equal  to  that  of  the  numerator.  In  other 
words,  if  a  positive  unit  radius  is  taken  as  the  denominator,  the 
length  and  sign  of  the  numerator  will  represent  the  function  in 
magnitude  and  sign.  We  have,  then,  simply  to  select  six  lines 
whose  ratios  to  the  radius  agree  with  the  definitions  of  Art.  54. 
The  ratio  of  the  subtended  arc  to  the  radius  is,  by  Art.  52,  the 
circular  measure  of  the  angle. 

Suppose,  then,  a  circle  of  unit  radius  drawn  with  its  center  at 
the  origin  of  coordinates. 

The  angle  is  placed  upon  the  axes  just  as  in  Art.  6,  and  from 
the  point  P  of  intersection  of  the  terminal  line  with  the  circle, 
perpendiculars  MP  and  NP  are  drawn  to  the  two  axes.  From 
the  two  points  Jl  and  ^  where  the  positive  axes  cut  the  circle, 
tangents  ^2^  and  BS  are  drawn  meeting  the  terminal  line  (pro- 
duced if  necessary)  in  the  points  T  and  S. 

Since  P,  T,  jS,  Figs.  60-63,  lie  in  the  terminal  line,  we  have,  at 
once,  in  accordance  with  Art.  54  (or  Art.  6) : 


MP  NP 

AT  ,        BS 

tan  a  =  -—- ,  cot  «  =  -— , 

OA  OB 


OT  OS 

sec«=^,         csc«  =  -^. 

But  by  construction, 

OP=OA=OB  =  1. 


LINE   REPRESENTATIONS 


81 


These  denominators  may  then  be  suppressed  and  the  functions 
represented  graphically  as  indicated  below : 


t 

\^ 

B 

^ 

\ 

\ 

A      , 

1      M 

1 
\        1 

^•\r 

Fig.  60 


Fig.  61. 


Fig.  62.  Fia.  63. 

sin  a  —  MP,        cos  a  =  NP, 

tan  a  =  AT,         cot  a  =  BS, 

sec  a=  OT,         CSC  a  =  OS. 

Moreover,  the  angle,  in  radians,  is  represented  as  follows 

arc  AP 


OP 


arc  AP. 


According  to  the  modern  view,  the  line  is  not  the  function,  but 
by  its  length  and  direction  represents  the  function  in  magnitude 
and  sign. 

Note  that  the  line  representing  the  tangent  is  always  drawn 
from  the  point  A  and  that  representing  the  cotangent  from  B. 
All  the  lines  are  read  from  the  axes  to  the  terminal  line.  Hori- 
zontal lines  are  positive  toward  the  right,  negative  toward  the  left. 
Vertical  lines  are  positive  upward,  negative  downward. 


82  THE   GENERAL   ANGLE 

By  means  of  the  Pythagorean  proposition,  and  the  theorems 
concerning  similar  triangles,  the  fundamental  relations  given  in 
the  preceding  article,  as  also  the  limitations  of  value  stated  in  Art. 
55,  are  readily  established.  So,  also,  the  subsequent  theorems  of 
trigonometry  may  be  interpreted  by  means  of  the  line  represen- 
tation of  the  trigonometric  functions.  This  graphic  interpretation 
frequently  presents  special  advantages.  This  is  the  case,  for  ex- 
ample, in  the  investigation  of  the  variation  of  the  functions  con- 
sidered in  Art.  5Q.  So,  too,  the  construction  of  the  graphs  of  the 
functions  as  treated  in  Art.  57  is  facilitated,  since  the  lengths  of 
the  defining  lines  may  be  transferred  by  the  use  of  dividers. 

EXERCISE    XVil 
Find  the  values  of  the  following  expressions  : 

1.  cos^  a  —  sin^  a,  when  a  =  arctan  (  —  |),  in  the  2d  quadrant. 

2.    •  H ,  when  a  =  sec-i(—  3),  in  the  3d  quadrant. 

1  —  tan  a     1  —  cot  ct 

3.    '- -,  when  a  =  arcsin(—  14),  in  the  4th- quadrant. 

tana  -  sec  a  +  1  \      o^/  ^ 


4. 


1 ,  when  a  =  cos-^M,  in  the  4th  quadrant. 

CSC  a  —  cot  a     esc  a  +  cot  a 

Solve  the  following  equations,  finding  all  the  angles  less  than 
2  TT  that  satisfy  each  equation  : 

5.  cos/?  =^. 

6.  tan  y8  =  -  Vs. 

7.  sin  2  a  =  -  ^V^. 

8.  cot  3  a  =:  1. 

9.  4  sin2  a  —  4  cos  a  —  1  =  0. 

10.  3tan2y8-l  =  0. 

11.  2  sin  ^  cos  ^  -  sin  ^  =  0. 

12.  2  sin  a  +  \/3  tan  a  =0. 

In  exercises  13-24,  verify  the  given  identities  by  transforming 
the  first  member  into  the  second. 

13.  (sin  a  +  cos  a)  (cot  a  +  tan  a)  =  sec  «  +  esc  a. 

14.  (sec  a  —  cos  a)  (esc  a  —  sin  a)  =  sin  a  cos  a. 

15.  ^^"^  +  ^^^^^tan«cot)8. 
cot  a  +  tan  ft  ' 


PERIODICITY  83 

16.    (r  cos  Oy  -{-  (r  sin  0  cos  <^)2  +  (r  sin  ^  sin  <f>y  =  r\ 

^_      tan  a  —  tan  y8     ^  cot  a  cot  ^  +  1  _  ^ 
1  +  tan  a  tan  ^        cot  y8  —  cot  a 

18.  CSC  a  (sec  a  —  1)  —  cot  a  (1  —  cos  a)  =  tan  a  —  sin  a. 

19.  (sin  a  cos  13  —  cos  a  sin  ^)^  +  (cos  a  cos  ^  +  sin  a  sin  ^)2  =  1. 

20.  sec  ct  CSC  a  (1  —  2  cos'^  a)  +  cot  a  =  tan  a. 

21.  (sin  a  cos  /?  +  cos  «  sin  f^)'^  +  (cos  ot  cos  /?  —  sin  a  sin  I3y  =  l. 

oo  o  2         (l-tan2a)2       .  » 

22.  sec-  a  csc^  ct  —  ^^ ^  =  4. 

tan-^  a 

23.  (cos  a  +  V—  1  sin  a) (cos  a  —  V  —  1  sin  a)  =  1. 

24.  (cos  a  +  V—  1  sin  a)^  +  (cos  a  —  V—  1  sin  «)2  z=  4  cos^  a  —  2. 

25.  By  means  of  Fig.  60  show  that,  when  0  is  acute  and  measured  in 
radians,  sec  0  >  tan  0>0>  sin  0. 

26.  By  means  of  Fig.  60  show  that,  when  6  is  acute  and  measured  in 
radians,  esc  ^  >  cot  ^  >  ( '^  —  ^  j  >  cos  ^. 

61.  Periodicity  of  the  trigonometric  functions.  It  was  pointed 
out,  in  Art.  53,  that  if  two  angles  differing  by  an  integral  multiple 
of  360°  are  placed  on  the  axes,  their  terminal  lines  coincide.  As 
an  immediate  consequence,  it  follows  that  corresponding  functions 
of  the  two  angles  are  identical.     Thus  we  may  write 

sin  (2  kir  -^  a)  =  sin  a, 
and,  in  general, 

F(2kiT  -\-  a)  =  jP(a), 

where  F  denotes  the  same  function  in  both  members  of  the  equa- 
tion, and  k  is  an  integer. 


62.    Functions  of 


k^±a  .     Precisely  as  in  Art.  10,  37,  and 

38,  we  may  express  the  functions  of  the  angles  ±  a,  90°  ±  a, 
180°  ±  a,  270°  ±  a,  360°  ±  a,  and  other  similarly  compounded  angles 
in  terms  of  the  functions  of  cc,  no  matter  what  the  quadrant  of  the 
angle  a.  Because  of  the  periodicity  brought  out  in  the  preced- 
ing article,  it  is  not  necessary  to  carry  the  investigation  beyond 
the  five  multiples  of  the  right  angle  mentioned ;  indeed,  the  fifth 
reduces  to  the  first.  On  account  of  the  double  signs  and  the  pos- 
sibility^ of  a  belonging  to  any  one  of  the  four  quadrants,  there 
exist  thirty-two  distinct  cases.     The  demonstration  is  tlie  same 


84 


THE   GENERAL   ANGLE 


for  all  cases,  involving  the  same  proportionality  of  sides  of  similar 
triangles  and  the  same  question  of  agreement  or  opposition  of  signs. 
The  working  out  of  the  proof  in  three  characteristic  instances 
should  be  sufficient  to  enable  the  student  to  do  the  same  for  any 
and  all  cases.  The  theorem  is,  however,  somewhat  elusive,  and 
the  student  can  completely  master  it  and  render  it  an  infallible 
instrument  only  by  actual  careful  construction  and  proof  of  most 
of  the  cases.  Upon  first  study  it  may  be  well  to  limit  considera- 
tion to  the  cases  in  which  a  is  of  the  first  quadrant. 

Let  it  be  required  first  to  express  the  functions  of  (180°  -f-  a) 
in  terms  of  functions  of  a,  when  a  is  an  angle  of  the  first  quad- 
rant. If,  in  Fig.  64,  Z  XOA  =  a, 
thenZXOB=/3=lS0°-\-a.  The 
two  triangles  OMP  and  OJSfQ  are 
similar,  the  pairs  of  correspond- 
ing sides  being  v  and  v',  x  and  x\ 
and  y  and  y.  Notice  also  that 
x^  and  ?/'  are  negative,  all  the 
other  sides  being  positive.  Giv- 
ing due  attention  to  signs,  we 
Fig  64.  may   write  : 


sin  (180°  + a)  =  sin /3  = 


y 


sin  a, 


cos  (180°  + a)  =  cos /5  = 


=  —  cos  OJ, 

V 


tan  (180°+  a)  =tan  /5  =  ^  =  ^  =  tan  a, 


cot  (180°  +  a)  =  cot  /3  =  -  =  -  =  cot «, 

y     y 


sec  (180°+ a)  =  sec /? 


sec  a, 


esc (180°  +  «)  =  CSC  /3  =  ^ 


y 


-  =  —  CSC  a. 

y 


Again,  let  it  be  required  to  express  the  functions  of  (270®  — a) 
in  terms  of  functions  of  «,  when  a  is  of  the  first  quadrant.  In 
Fig.  Qb,  Z  XOA  =  «.  Z  XOB  =  /3  =  270°  -  a.  The  two  triangles 
OMP  and  ONQ  are  similar,  the  pairs  of  corresponding  sides  now 


FUNCTIONS   OF 


[..|±«] 


being  v  and  v\  x  and  y\  and  y  and  x\     The  sides  x^  and 
negative,  all  the  others  positive.     We  may  then  write; 


sin(270°-«)  =  sinyS  = 
cos(270°-a)=cos)8  = 
tan(270°-«)=tan^  = 
cot(270°-«)=coty8  = 
sec(270°-«)=secy8  = 
csc(270°-a)  =  csc,/S  = 


As  a  third  and  especially  important 
instance,  let  us  find  the  functions  of  —  a, 
when   a  is  of  the  second  quadrant.     In 
Fig.    m,    XOA  =  «,    XOB=  /3=  -  a. 
The  two  triangles  OMP  and  ONQ 
are  similar,  the  pairs  of  correspond- 
ing sides  being  v  and  v',  x  and  x\ 
y  and   y\  while   a;,  a;',   and  ^'  are    ^ 
negative. 

We  then  have,  as  before :  p^^  ^g 

sin (  —  a )  =  sin  ,8  =  ^  =  — ^=  —  sin  a, 


85 


are 


y 

v' 

- 

X 

V  ~ 

-  COS  a. 

x' 

v' 

- 

V 

-  sin  a. 

y' 

x' 

x 

y 

=  cot 

a. 

x' 

y' 

= 

y 

X 

=  tan 

a. 

v[ 



_ 

V  _ 

-  CSC  a. 

x' 

y 

v' 

y' 

= 

— 

V  _ 
X 

•  sec  a. 

X  X 

cos  (  —  a)  =  COS  ^  =  —  =  -  =  cos  a, 

v'       V 


tan  (  —  a)  =  tan  /9  =  '-^ 


cot  (  —  a)  =  cot  yS  =  —  = 


tana. 
—  cot  a, 


sec  (  —  a )  =  sec  ytj  =  —  =  -  =  sec  a, 


CSC  (  —  a)  =  CSC  /3  =  -r  = =  —  esc  a. 

v'  X 


86  THE   GENERAL   ANGLE 

It  will  be  noticed  that  whenever  the  number  of  right  angles 
involved  is  even  the  pairs  of  corresponding  sides  are  v  and  v\x  and 
x' ^  y  and  y^  ;  while  whenever  the  number  of  right  angles  is  odd 
the  pairs  of  corresponding  sides  are  v  and  v\  x  and  y\  y  and  x' . 
Thus  we  have  the  theorem  :  Any  function  of  an  even  number  of 

right  angles  plus  or  minus  a  is 
numerically  equal  to  the  same  func- 
tion of  a;  any  function  of  an  odd 
number  of  right  angles  plus  or  minus 
a  is  numerically  equal  to  the  cor- 
responding co-function  of  a;  the 
agreement  or  opposition  of  signs  is 
to  be  determined  from  the  quadrants 
of  a  and  of  the  compound  angle.  It 
may  easily  be  verified  that  in  all 
cases  this  agreement  or  opposition 
of  signs  is  the  same  as  when  a  is  of  first  quadrant. 

The  general  theorem  may  also  be  stated  as  follows  :  If  the  sum 
or  difference  of  two  angles  is  an  even  number  of  right  angles^  the 
functions  of  the  one  are  numerically  equal  to  the  same  functions  of 
the  other.  If  the  sum  or  difference  of  two  angles  is  an  odd  number 
of  right  angles^  the  functions  of  the  one  are  numerically  equal  to  the 
corresponding  co-functio7is  of  the  other.  The  agreement  or  opposition 
of  signs  is  to  be  determined  from  the  quadrants  of  the  two  angles. 

The  significance  of  the  theorem  is  made  clear  by  application 
to  an  example:  Required  to  find  the  value  of  cos  (810°  + a). 
Here  810°  =  9  x  90°,  an  odd  number  of  right  angles.  When  a  is 
considered  as  of  the  first  quadrant  (and  its  functions  consequently 
positive),  the  compound  angle  (810°  +  a)  is  of  the  second  quadrant 
and  hence  its  cosine  is  negative.  The  required  relation  is,  there- 
fore, 

cos  (810°  +  a)  =  —  sin  «, 

which  holds  for  all  values  of  a. 

Again,  to  find  the  value  of  tan  1230°.     We  have 

1230°  =  14  X  90°  -  30°,  and  is  of  second  quadrant. 

Then  tan  1230°  =  _  tan  30°  =  -  — . 

V3 

The  student  may,  if  he  prefers,  construct  the  figure  and  proceed 
as  in  the  demonstration  just  given. 


FUNCTIONS   OF    Fa;  •  |  ±  a"]  87 

As  a  consequence  of  these  relations,  it  follows  that  to  every 
inverse  function  correspond  two  angles,  lying  between  0  and  2  tt. 

Thus  arc  sin  a  =  a  and     ir  —  a^ 

arc  cos  h  =  a  and  2  tt  —  a, 

arc  tan  c=  a  and     tt  +  a, 

arc  cot d=  a  and     tt  +  «, 

arc  sec  e  =  a  and  2  tt  —  «, 

arc  esc/  =  a  and     tt  —  a. 

These  statements  should  be  verified  by  the  student. 

EXERCISE  XVIII 
Express  in  terms  of  a  positive  angle  less  than  45° : 

1.  sin  700^  4.   cot  -  35°. 

2.  cos  260°.  5.    CSC  930". 

3.  tan436^  6.    sec  1400°. 

Find  the  value  of  cos  a  -}-  sin  a  and  of  tan  a  —  cot  a  when  a  has 
the  value 

7.    ?.  10.    '^'^. 

6  (j 

8.  -2j-.  11   iijr 

3  ■       3    * 

9.    1^.  12.    -^. 

4  4 

Find  all  the  values  between  0°  and  360°  of 

13.  arctaii  V3.  16.    arcsec  2. 

14.  csc-i(-  V2).  17.    arccot(-l). 

15.  arccos  (-  .5).  18.    sin-i  (-  ^  V3). 

Find  the  value  of 

19.  sin  480°  sin  690°  +  cos  (  -  420°)  cos  600°. 

20.  tan  840°  cot  420°  +  tan  (-  300°)  cot  (-  120°). 

21.  tan  llr  tan  ii5  +  cot  (  -  11^)  cot  (  -  t'^) . 

22.  sin  1|^  cos  (  -  ^)  -  sin  If  cos  (  -  ^ 


88  THE  GENERAL   ANGLE 

23.  If  sin  200°  30'  =  -  .35,  find  cos  830°  30'. 

24.  If  tan  558°  26'  =  i,  find  cot  468°  26'. 

25.  If  cot  520°  =  -  a,  find  sin  160°. 

26.  If  cos  590°  =  -  m,  find  tan  850°. 

27.  Express  cos  (a  —  90°)  as  a  function  of  a. 

28.  Express  sin  (a  —  180°)  as  a  function  of  a. 

29.  Express  tan  (a  —  360°)  as  a  function  of  a. 

30.  Express  cot  (a  —  270)°  as  a  function  of  a. 


CHAPTER   VIII 

FUNCTIONS  OF   TWO  ANGLES 

63.  Formulas  for  sin  (a  +  p)  and  cos  (a4-  p).  Suppose  a  and 
y8  to  be  acute  angles.  In  Fig.  67  (ct  +  /3)  is  acute ;  in  Fig.  68 
(a  +  yS)  is  obtuse.  The  following  demonstration  applies  to  both 
figures. 

Let  ZXOA  =  a,  ZAOB  =  0',  then  ZX0B  =  a  +  /3.  From 
P,  a  point  in  0J5,  draw  Pil[f  perpendicular  to  OX,  PQ  perpendic- 


M  N 


Fig.  67. 


>X 


Fig.  68. 


ular  to  OA^  and  from  Q  draw  §iV  perpendicular  to  OX,  and  QR 
perpendicular  to  MP.  The  angle  RPQ  ==  a  and  PP  =  QP  cos  a, 
PQ  =z  QP  sin  a,  by  Art.  J6.     By  the  same  article. 

MP=OP  s'm  (a  +  /3). 
Also  MP  =  MR  +  RP  =  NQ-\-RP 

=  0§  sin  a  +  ^jP  cos  a 

=  OP  sin  a  cos  yS  +  OP  cos  a  sin  y8. 

Equating  the  two  values  of  MP  and  dividing  through  by  the 
common  factor  OP,  we  have  the  theorem 


sin  (a  +  p)  =  sin  a  cos  P  +  cos  a  sin  p. 

80 


0) 


90  FUNCTIONS   OF   TWO   ANGLES 

In  like  manner 

Oi)^f=  OP  cos  («+)S), 
and  also  OM=^ON-MN=  ON- RQ 

=  OQ  cos  a  —  QP  sin  a 
,  =  OP  cos  a  cos  /S  —  OP  sin  a  sin  p. 

Hence  the  companion  theorem 

cos  (a  +  p)  =  cos  a  cos  p  —  sin  a  sin  p.  (2) 

These  are  called  the  addition  formulas  and  are  fundamental 
in  trigonometry. 

64.  Extension  of  addition  formulas.  The  two  formulas  of  the 
last  article  were  proved  only  for  angles  both  of  the  first  quadrant. 
It  remains  to  be  shown  that  they  hold  when  a  and  /S  denote  any 
angles. 

First,  let  a  be  an  angle  of  the  second  quadrant.  Then 
(^  (=  a  —  90°)  is  an  angle  of  the  first  quadrant.  Now  a  =  90° +  <^, 
so  tliat,  by  Art.  38, 

sin  a  =  cos  <^,  cos  a  =  —  sin  <f). 

Since  <^  is  of  the  first  quadrant,  the  formulas  of  Art.  63  apply 
and  we  have 

sin  (a H- yS)  =  sin  (90°  +(/)+yS) 

=:COS(</>  +  /S) 

=  cos  (^  COS  /3  —  sin  <f)  sin  /3, 
=  sin  a  cos  /3  +  cos  a  sin  /3. 
Likewise 


cos  («  +  /8)  =  cos  (90°  +  (^  +  /3) 
=  -sin(^  +  /3) 
=  —  sin  </)  cos  /3  —  cos  <f>  sin  /S, 
=  cos  a  cos  yS  —  sin  a  sin  y8. 

The  formulas  are  therefore  true  when  one  angle  is  of  the  first 
and  the  other  of  the  second  quadrant.  By  adding  90°  succes- 
sively to  each  of  the  angles,  the  formulas  are  established  for  two 
positive  angles  of  all  quadrants.  If  one  of  the  angles  is  negative, 
it  can  be  augmented  by  such  an  integral  multiple  of  360°  as  to 
produce  a  positive  angle  possessing  the  same  functions. 

The  addition  formulas  are,  therefore,  true  for  angles  of  any 
size. 


ADDITION   AND   SUBTRACTION   FORMULAS  91 

EXERCISE  XIX 

Evaluate  the  addition  formulas  for 

1.  a  =  60°,    /?  =  30°.  3.    a  =  240°,  13  =  150°. 

2.  a  =  45°,    /?  =  90°.  4.    a  =  300°,  ^  =  150°.  . 

5.  «  =  arctanf,  ^  =  arccos  (—  -^j),  a  first  quadrant,  ft  second. 

6.  a  =    sin-i  (—  j^^),  /3  =  cot-^  5^,  a  fourth  quadrant,  /?  third. 

Find  the  value  of 

7.  cos  (^  +  a)  cos  (1  +  /«)  -  sill  (|  +  «)  sin  (|  +  )»). 

8.  sin  (7  + a)  cos(^+/8)  +  cos  (|  +  a)  sin  (|  +  ysV 

9.  sin  (1  +  n)  a  cos  (1  -  n)  «  +  cos  (1  +  n)  «  sin  (1  —  n)  a. 

10.  cos  (1  +  n)  a  cos  (1  —  n)  a  —  sin  (1  +  n)  a  sin  (1  —  n)  a. 

11.  sin  (^  +  <^)  cos  {6  -  <f>)  +  cos  (^  +  <^)  sin  (^  -  <^). 

12.  cos  (^  —  <^)  cos  ^  —  sin  (^  —  <^)  sin  <^. 

13.  Evaluate   the  addition  formulas  for  a  =  60°,  ^  =  45°,  and  thus  find 
sin  105°,  cos  105°,  sin  15°,  cos  15°. 

14.  Evaluate  the    addition  formulas   for  a  =  45°,  ^  =  30°,  and  thus  find 
sin  75°,  cos  75°,  sin  15°,  cos  15°. 

65.    Subtraction  formulas.     In  the  addition  formulas  replace  /3 
by  —  fi.     We  have 

sin  (a  —  yg)  =  sin  a  cos  (—/?)  +  cos  a  sin  (—  yS). 
But  by  Art.  62, 

sin  (—  y8)  =  —  sin  y5,  cos  (—  jS')  =  cos  y5. 
Making  this  substitution,  we  have 

sin  (a  —  P)  =  sin  a  cos  p  —  cos  a  sin  p.  (1) 

In  like  manner 

cos  (a  —  yS)  =  cos  a  cos  (—  yS)  —  sin  a  sin  (—  yS), 
or,  by  the  same  substitution, 

cos  (a  —  p)  =  cos  a  cos  p  +  sin  a  sin  p.  (2) 


92  FUNCTIONS   OF   TWO   ANGLES 

66.    Formulas  for  tan  (a  ±  p),  cot  (a  ±  p).     From  Arts.  59  and 

63  we  have 

.       .     ,    a\      sin  (a  4-  yS) 

tan(a  +  /3)= )    ^  ^( 

cos  («  +  /3) 

_  sin  a  cos  /8  +  cos  a  sin  /3 
cos  a  cos  yS  —  sin  a  sin  yS 

sin  a  cos  /3      cos  «  sin  ^ 
cos  a  cos  /8      cos  a  cos  /S 


cos  a  cos  yS  _  sin  a  sin  /S 
cos  ct  cos  ^      cos  ct  cos  ff 

or,  finally, 

-I      ^     .    ON       tan  a  +  tan  B  ^^  ^ 

tan  (a  +  P)  =  ;; ^  •  (1) 

^         ^^      1-tanatanP  ^  ^ 

In  like  manner  we  may  derive 

X      /         o\        tan  a  —  tan  B  ^^;. 

tan  (a  —  P)  = -^-  (2) 

^^      1  +  tanatanP  ^  ^ 

cot(«  +  ^)=22^fi±|l, 

Sin  (ct  +  yS) 

_  COS  a  cos  /6  —  sin  a  sin  /3 
sin  a  cos  fi  +  cos  a  sin  yS' 

cos  ct  cos  /3  sin  cc  sin  y8 

__  sin  a  sin  y8  sin  a  sin  ^ 

sin  a  cos  ^5  cos  a  sin  yS' 

sin  a  sin  jS  sin  a  sin  /3 


Again, 


or 


Likewise 


^         ^^       cot  p  + cot  a  ^  ^ 

J.  r         ox       cot  a  cot  p  +  1  ^ . ^ 

cot  (a  —  P)  = ~ ^ — -  •  (4) 

^         cot  p  -  cot  a  ^  ^ 

EXERCISE  XX 

1.  Demonstrate  geometrically  the  formula  for  sin  («  —  y8),  when  a  >  (3^ 
both  acute. 

2.  Demonstrate  geometrically  the  formula  for  cos  (a  —  (3),  when  a  >  /?, 
both  acute. 


FUNCTIONS   OF  2  a  AND   ^  93 

Evaluate  the  formulas  of  Art.  ^^  for 

3.  a  =  60°,  y8  =  120°.  5.   a  =  arcsin  ^V,  ^  =  arctan  -  ^f . 

4.  a  =  240°,  y8  =  150°.  6.   a  =  cot-i  |^,  ^  =  cos-i  ||. 

Evaluate  the  formulas  of  Art.  ^^  for 
7.    a  =  330^  ^  =  150°.  8.   a  =  210°,  )8  =  300°. 

9.   a  =  cos-i  y5_,  ^  :^  tan-i  (  -  f ). 

10.  a  -  arccot  \\,  /?  =  arctan  f  Q. 

11.  Find  the  functions  of  15°  by  putting  a  =  45°,  (3  =  30°. 

12.  Find  the  functions  of  15°  by  putting  a  =  60°,  /S  =  45°. 

Show  that 

13.  sin  (a  +  (3)  sin  (a-  /3)  =  sin-^ a  —  sin^ (3  =  cos^  /?  —  cos^ a. 

14.  cos  (a  +  /3)  cos  (a  -  yS)  =  cos^  a  —  sin^  ^  =:  cos^  (3  —  sin^  a. 

Expand  by  successive  applications  of  the  formulas : 

15.  sin(«  +  /?+y).  17.    tan(«  +  ^  +  y). 

16.  cos(a  +  j8  +  y).  18.    cot(a+^  +  y). 

Show  that 

19.    sin(^+ «  J  -  sin  f  "^  -  a  j  =  sina. 

(  -  —  a  y  =  \/3  cos  a. 
6  ■      /    ■  \6        / 

67.  Functions  of  twice  an  angle.     In  the  addition  formulas  of 
Arts.  63  and  6Q,  place  jS  =  a.     We  then  obtain 

sin  2  a  =  2  sin  a  cos  a,  (1) 

cos  2  a  =  cos^  a  —  sin^  a,  (2) 

=  l-2sin2a,  (2  a) 

=  2cos2a-l.  (2?0 

.     n  2  tan  a  ^ox 

tan2a  =  - —- (3) 

1  —  tan^  a 

.  o         cot^  a  —  1  ^  ,  X 

cot  2  a  =    ~ (4) 

2  cot  a 

68.  Functions  of  half  an  angle.     From  Art.  67  we  may  write 

cos  2  /3  =  1  -  2  sin2  /8, 


94  FUNCTIONS   OF   TWO   ANGLES 

and  solving  for  sin/3, 

sin  ^  =  Vi(l-cos2yS). 

Now  placing  2  /S  =  a,  so  that  y^  =  ^,  we  obtain 


sin  I  a  =  Vi  (1  -  cos  a).  (1) 

Similarly 

cos  2  yS  =  2  cos2  y8  -  1 ; 
so  that 

cos  y8  =  Vi(l  +  COs2y8), 

and,  with  the  same  substitution, 

cos  I  a  =  Vi(l  +  cosa) .  (2) 

Dividing  the  first  formula  by  the  second,  we  get 

.      1  ^  /I  —  cos  a  >-Q\ 

taii-a  =  ^— ,  (d) 

2  ^  1  +  cos  a 

and  inverting, 


.1         ^  /I  +  COS  a  ^^N 

cot     a  =  X|- (4) 

a  ^  1  —  cos  a 

Rationalizing  the  numerators  of  the  last  two  formulas,  we  get 

other  useful  forms, 

^-^^  1        1  —  cos  a  ,r>, 

tan  -a  = -. ,  (5) 

2  since 

^^i.  1         1 4-  cos  a  y^«>, 

cot  -  a  =  — : (6) 

2  sin  a 

Query.  —  Why  are  formulas  (1)  to  (4)  ambiguous  in  sign,  while  (5)  and  ((>) 
are  apparently  not  ? 

EXERCISE  XXI 
Find  the  values  of 

1.  The  functions  of  60°  from  those  of  30°. 

2.  The  functions  of  120°  from  those  of  60°. 

3.  The  functions  of  75°  from  those  of  150°. 
«    4.  The  functions  of  15°  from  those  of  30°. 

Find  the  values  of  the  functions  of 

5.  2  arctan  :f%.  8.  I  arctan  \\l, 

6.  2  cos-i  (—  j\).  9.  arcsin  ^^  +  2arccot  f. 

7.  J  sin-i  (  —  M)'  ^0-  arctan  ^^  —  2  arccos  f . 


CONVERSION   FORMULAS  95 


Transform  the  first  member  into  the  second : 

11.  l+sin^-cos2^^^^^^^ 

cos  ^  +  sin  2  ^ 

12.  l+cos^  +  cos2^^^^^^ 

sin  6  +  sin  2  6 


13.    (VT+sina+ Vl  —  sin  a)2z=4cos2ia. 


14.  ( Vl  +  sin  a  —  Vl  —sin  a) 2  =  4 sin^ i  a. 

15.  tan  f  -  +  a  J  -  tan  f  -  -  a  J  =  2  tan  2  a. 

16.  cot  [-  +  a^  -  cot  [  -  -  a]  =  -  2  tan  2  a. 

Find  the  values  of  a  which  satisfy  the  following  equations 

17.  (2  + V3)(l-sin2a) -2cos22a  =  0. 

18.  sin  2  a  +  2  cos  2  a  =  1. 

19.  4  sec2  2  a  +  tan  2a  =  l. 

20.  CSC  2  a  +  cot  2  a  =  2. 
Show  that 

21.  tan-i  ^^^  =  cos-i  ^ 


3  Va;2  -  4  a:  +  13 

2  a:  + 

— — ^i^:^^::^^::::^  —  arccsc  — --- 

y/x'^  +  2  x  -  3  2 


9  a:  + 1 

22.   arctan  —        " —  arccsc  - 


23.    Find  sin  (^-  2  tan-i 


'\-- 


\-x\ 
l+xj' 


24.  Find  sin  fsin-i  m  4-  tan"!^^-^^ — ^V 

\  ml 

25.  Find  sin  f  arccos  (1  —  a)  —  2  arctan  -^  — ^^ —  |  • 

26.  Find  cos  (arccos  (1  —  2  a)  —  2  arcsin  Va). 

69.    Conversion  formulas  for  products.     Adding  the  two  first 
formulas  of  Arts.  63  and  65,  we  have 

sin  («  +  ,5)  +  sin  («  —  yS)  =  2  sin  a  cos  /3, 

or,  reversing  and  dividing  by  2, 

sin  a  cos  p  =  1  [sin  (a  +  p)  +  sin  (a  -  p)] .  (1) 

If  we  subtract,  instead  of  adding,  we  get 

sin  (a  +  yS)  —  sin  (a  —  /S)  =  2  cos  a  sin  /3, 
or 

cos  a  sin  p  =  i  [sin  (a  +  P)  -  sin  (a  -  p)].  (2) 


96  FUNCTIONS   OF   TWO   ANGLES 

Treating  the  two  second  formulas  in  like  manner,  we  obtain 

cos  a  COS  p  =1  [cos  (a  +  p)  +  cos  (a  —  p)],  (3) 

and 

smasmp  =  -l  [cos(a  +  p) -cos(a- P)].  (4) 

By  means  of  these  formulas,  products  of  sines  and  cosines  are 
expressed  as  sums  or  differences.  By  successive  applications 
higher  powers  and  products  are  reducible  to  expressions  linear 
in  sines  and  cosines.  The  same  transformations  may  often  be 
effected  by  application  of  the  formulas  of  Art.  67,  written  in  the 
form 

sin  a  cos  a  =  |  sin  2  a,  (5) 

sin^  a  =  J  (1  —  cos  2  a),  (6) 

cos2a=  I  (l  +  cos2a).  (7) 

EXERCISE   XXII 

Reduce  the  following  products  to  linear  expressions : 

1.  sin  5  a  cos  S  (^.  6.    sin  a  cos^  u- 

2.  cos  6  a  sin  4  cc.  7.   cos^  a. 

3.  sin  7  a  sin  3  a.  8.   sin^  a. 

4.  cos  2  a  cos  5  a.  9.   cos^  a  sin^  a. 

5.  sin^  a  cos  a.  10.    sin^  a  cos^  a. 

Show  that 

11.  cos  a  sin  (/?  —  y)  +  cos  /S  sin  (y  —  a)  +  cos  y  sin  («  —  ^)  =  0. 

12.  sin(^  -  y)  sin  («  -  8)  +  sin(y  -  a)  sin  (/3  -  8)  +  sin  (a-/3)  sin  (y- 8)  =  0. 

13.  sin  — cos^+  sin^cosi^  =  0. 

5  5  5  5 

14.  2  cos^cos^  +  sin^  +  cos^  =  0. 

8         4  8  8 

Solve  for  a,  making  use  of  Art.  10. 

15.  COS  (50'^  +  a)  sin  (50°  -a)-  cos  (40°  -  a)  sin  (40°  +  a)  =  0. 

16.  sin  (70°  +  a)  sin  (70°  -  a)  +  sin  (20°  +  a)  sin  (20°  -  a)  =  0. 

Solve  for  a,  making  use  of  Art.  69. 

17.  cos  3  a  +  cos  9  a  =  0. 

18.  sin  5  a  —  sin  10  a  =  0. 


CONVERSION   FORMULAS  97 

19.  cos  (a  +  0)  cos  (a-  6)  +  cos  (3  a  +  6)  cos  (3  a  -  6)  =  cos  2  0. 

20.  sin  (a  +  6)  cos  (a  -  ^)  +  sin  (3  a  +  ^)  cos  (3  ct  -  ^)  =  sin  2  0. 

70.    Conversion   formulas   for    sums  and    differences.     In   the 

process  of  deriving  the  formulas  of  the  last  article,  before  revers- 
ing and  dividing  by  2,  substitute    a  +  /3  =  ^,  a— (3  =  6^  so  that 

We  then  obtain  the  following  formulas : 

sincj)  +  sin  6  =  2  sin^        ^^^     T    '>  (^) 

sine))- siii6  =2cos^        sin^       ,  (2) 

coscf) +  cos6  =2  cos^-— —  cos^— — ,  (3) 

cos<))  — cosG  =  -  2sin^-- — ^i^^^^ —  (^) 

These  formulas  serve  to  effect  transformations  converse  to 
those  mentioned  in  Art.  69. 

71.    Multiple  angles.    In  the  formula  for  sin  (a  +  /3)  put  y8  =  2  a. 

Then 

sin  3  a  =  sin  a  cos  2  a  +  cos  a  sin  2  a 

=  sin  a  —  2  sin^  ce  +  2  sin  a  cos^  a 

=  3  sin  a  —  4  sin"^  a. 

Again,  cos  3  a  =  cos  a  cos  2  a  —  sin  a  sin  2  a 

=  2  cos^  a  —  cos  a  —  2  sin^  «  cos  a 

=  4  cos^a  —  3  cos  a. 

In  like  manner  the  other  functions  of  3  ct  and,  by  repeating  the 
process,  the  functions  of  any  integral  multiple  of  a  may  be  ex- 
pressed in  terms  of  functions  of  a. 

EXERCISE    XXIII 

Show  that 

1.   ?i5A«±^iEij?  =  tan5a. 
cos  6  a  +  cos  4  a 

2    co^3^t-_cos^^^^^^^ 
sin  3  a  +  sin  5  a 


98  FUNCTIONS  OF   TWO  ANGLES 


«     sin  7  «  —  sin  5  a      , 

3. =  tan  a. 

cos  7  a  +  cos  5  a 

.     cos  4  a  —  cos  2  «         j.      o 

4.  - — : — -—  =  -  tan  3  a. 

sin  4  «  —  sm  2  cj 

5  sina-sin^^^^^^gM:^^^^a-^^ 
sin  ct  +  sin  |8  2  2 

6  cos«  +  co^^_^^^«+^^^^«-^ 
cos  a  —  cos  )8  2  2 

^    cos  3  ^  +  2  cos  5  ^  +  cos  7  0  _      ,  ^  ^ 
sin  3  ^  +  2  sin  5  ^  +  sin  7  ^  ~ 

8     sin  ^  -  2  sin  4  ^  +  sin  7  6  _.       .q 
'   cos  ^  -  2  cos  4  ^  +  cos  7  ^  ~ 


Solve  the  following  equations  : 

9.   cos  6  +  cos  5  ^  =  cos  3  ^. 

10.  sin  ^  + sin  5^  =  sin  3^. 

11.  sin  2  ^  +  2  sin  4  ^  +  sin  6  ^  =  0. 

12.  cos3^  +  2cos4^+cos5^  =  0. 

Derive  the  formulas  for : 

13.  cot  3  a.     (In  terms  of  cot  a.) 

14.  tan  3  a.     (In  terms  of  tan  a.) 

15.  sin  4  a. 

16.  cos  4  a. 

Solve  the  equations : 

17.  sin  3  a  =  V2  sin  2  a. 

18.  V3cos3a  +  2sin2a  =  0. 

19.  cos  3  a  =  cos  a  cos  2  a. 

20.  sin  3  a  =  sin  a  cos  2  a. 


CHAPTER  IX 


ANALYTIC  TRIGONOMETRY 


The  foregoing  chapters  constitute  an  introduction  to  the  elementary  principles 
of  trigonometry.  The  student  ought  now  to  be  prepared  for  a  more  advanced 
study  of  the  theory  of  the  trigonometric  functions,  which  may  be  entitled  analytic 
trigonometry.  It  is  beyond  the  scope  of  this  book  to  consider  more  than  a  few  of 
the  most  important  topics  which  might  be  discussed  under  this  head.  For  a  more 
extended  treatment  the  student  is  referred  to  the  treatises  by  Henrici  and  Treut- 
lein,  Hobson,  Lock,  Loney,  Todhunter,  and  others,  and,  of  course,  to  articles  in 
the  various  mathematical  journals. 

72.  Limits  of  6/sin  6  and  6/tan  9  as 
6  approaches  zero.  Let  6  be  an  acute 
angle  measured  in  radians.  Con- 
struct, as  in  Fig.  69,  the  angle 
XOP  =  ^,  repeated  symmetrically 
as  XOQ.  Draw  through  P  the  arc 
PAQ  with  center  0,  the  chord  PMQ, 
and  the  broken  or  double  tangent 
PTQ,     Then 


AP 
OP 


=  6, 


MP 


OP 


=  sin  6^ 


TP 
OP 


=  tan  6, 


By  elementary  geometry, 

PMQ  <  PAQ  <  PTQ. 

Whence,  dividing  by  2  and  by  OP, 
sin  S  <  e  <  tan  6. 

Dividing  equation  (1)  through  by  sin  6,  wo  have 

e 


(1) 


1  < 


sin 


e 


<  sec  6. 


Now  in  Art.  12  it  was  proved  that  as  6  approaches  the  limit  0, 
cos  B  and  its  reciprocal  sec  6  approach  the  limit  1.  Thus,  the 
value  of  ^/sin  6  is  always  intermediate  between  1  and  a  number 
that  approaches  the  limit  1,  as  0  approaches  0.     The  ratio  ^/sin  6 


99 


100  ANALYTIC   TRIGONOMETRY 

must,  therefore,  approach  the  limit  1  at  the  same  time.     This  is 
expressed  symbolically  by  writing 

Again,  dividing  equation  (1)  through  by  tan  0,  we  get 
cos  e  <  — ^  <  1. 

tan  u 

Now  as  0  approaches  0,  cos  6  approaches  1,  and  hence,  as  before, 
O/tand  approaches  the  limit  1  at  the  same  time.     Symbolically, 

0=0  Vtan  0/ 

Note.  —  Since  sin  6  and  tan  d  both  approach  0  along  with  6,  it  might  seem  that 
they  therefore  approach  equality,  and  then  the  theorems  would  follow.  The  fallacy 
of  assuming  that  the  limiting  form  -  has  the  value  1  will  appear  on  considering  the 

following  instances.     The  circumference  and  area  of  a  circle  approach  zero  simul- 
taneously with  the  radius.     We  have,  however,  the  general  relations 

Circumference      2  Trr      ^  ^  ^  6.28318  ••., 


Radius  r 

Area   _  irr^ 
Radius       r 


=  Trr  =  3.14159  ...r. 


Now  when  r  approaches  the  limit  0,  the  limit  of  the  first  ratio  is  the  constant  2  tt, 
and  the  limit  of  the*second  ratio  is  0. 

The  limiting  form  -  will  be  discussed  at  length  in  calculus.  (See  Townsend 
and  Goodenough's  ''First  Course  in  Calculus,"  Art.  13.) 

Example.  If  0  is  increased  by  an  angle  S,  let  it  be  required 
to  determine  the  limit  of  the  ratio  of  the  consequent  increase  in 
sin  6  to  the  increment  3  of  6,  as  that  increment  8  approaches  zero. 
By  Art.  70,  we  have 

2cosf(9  +  |^sin  I 
sin  ((9 +  3) -sin  (9  _  V        27         2 


=  cos (6  + 


sin  - 
S\  2 


2/        8 
2 


De  moivrp:'S  theorem  lOi 

Now   when  B   approaches    0,    cos  ( ^  +  ^ )  approaches   cos  6  and 

.    8 

sm- 

— -—  approaches  1. 
o 

2 

Hence 

lim  sin  ((9  +  3)  -  sin  3  ^  ^^^  ^ 

6  =  0  g 

It  will  be  noticed  that  the  numerator  and  denominator  approach 
0  simultaneously,  but  that  the  limit  of  the  value  of  their  ratio  is 
a  number  somewhere  between  —  1  and  +  1,  and  depending  upon 
the  value  of  0. 

Examples 

In  like  manner  find  the  limits  as  3=0,  of 

,     cos  (6  +  8)  —  cos  0 

8 

n,     sec  (0  -\-  8)  —  sec  0  ^c<  -r«  •     j.  r       •      \ 

2.    ^^ — ^^—^ •        (Suggestion.     Express  in  terms  of  cosme.) 

«       CSC  (0  -\-S)   -  CSC  $ 

3. U 

4  tan  (0  +  8)  —  tan  0        (Suggestion.     Express  in  terms  of   sine  and 

8  cosine.) 

5  cot  (^  +  8)  -cot^ 

73.  De  Moivre's  theorem.  If  we  adopt  the  customary  nota- 
tion z  =  V—  1,  so  that  P  =  —1,  we  have,  on  performing  the  mul- 
tiplication, 

(cos  a+  i  sin  a)  (cos  /3  -f  ^  sin  /3)  =  cos  a  cos  /3  —  sin  a  sin  ^ 

+  {  (sin  a  cos  /3  4-  cos  a  sin  /3) 

=  cos  (« -1- /3)  +  «  sin  ((X  + /3),  (1) 

a  relation  which  holds  for  all  values  of  a  and  /3,  whether  positive 
or  negative. 

Putting  y8  =  a,  we  get 

(cos  a-\-i  sin  a)^  =  cos  2a  +  i  sin  2  a. 
Again,  putting  /3  =  2  a  in  equation  (1)  and  making  use  of  the 
relation  just  established,  we  get 

(cos  a+  i  sin  a)^  =  (cos  a-{-i  sin  cc)  (cos  2a  +  i  sin  2  a) 
=  cos  3  a  -f  i  sin  3  a. 


102  ANALYTIC   TRIGONOMETRY 

Repetition  of  this  process  proves  the  relation 

(cos  a  H-  ^  sin  a)"  =  cos  na  +  i  sin  na  (2) 

for  all  positive,  integral  values  of  n. 
It,  is  evident^  ypon  multiplying,  that 

''"''     rcr.«.    '(^QQs  ^_|_  ^- gii^  ^^(^cosa  — z  since)  =  1, 

whence 

(cos  a  4-  i  sin  a)~^  =  cos  a  —  i  sin  a. 

Suppose  71  to  be  a  negative  integer.  Let  n  =  —  m^  where  m  is 
a  positive  integer.     Now 

(cos  P  —  i  sin  /3)""'  =  (cos  yS  +  ^  sin  yS)"" 

=  cos  myS  +  ^  sin  m^. 

Substituting  m  =  —  n  and  yS  =  —  a,  we  get 

(cos  a  +  z  sin  «)"  =  cos  Tia  +  i  sin  Tia, 

true  also  for  negative  integral  values  of  n. 

Suppose  71  to  be  a  fraction,  either  positive  or  negative.     Let 

71  =  -,  where  r  and  s  are  integers.     Now 

r  1 

(cos  yS  +  z  sin  y8)  *  =  (cos  ryS  +  ^  sin  rp~) ' . 
Raising  both  members  to  the  sth  power, 

r 

(cos  Sy8  +  i  sin  s/3)  *  =  cos  r/3  +  i  sin  ry8. 

Introducing  -  =  ti,  and  putting  sfS  =  a,  so  that  r^=  -  -  s^=na, 
s  '  s        . 

we  get  (cos  a  +  ^  sin  a)"  =  cos  7i«  +  i  sin  7i«. 

This  relation,  therefore,  holds  for  all  rational  values  of  n. 
By  an  argument  involving  the  method  of  limits  it  can  be  proved 
also  for  all  irrational  values  of  n.  This  is  De  Moivre's  theorem,  an 
instrument  of  great  importance  in  some  branches  of  mathematics. 

Example.  An  illustration  of  its  use  is  afforded  by  applying 
it  to  tlie  derivation  of  the  formulas  for  the  sines  and  cosines  of 
multiple  angles.     Thus 

cos  3  a  -h  ^  sin  3  a  =  (cos  a  -f- 1  sin  a)^ 

=  cos^  a  4-  3  ^  cos^  «  sin  «  —  3  cos  a  sin^ a—  i  sin^  a. 


COMPLEX  NUMBERS  103 

On  equating  the  real  terms  on  each  side,  and  also  the  imagi- 
nary terms,  separately,  we  have  at  once 

cos  3  a  =  cos^  a  —  3  cos  a  sin^  a 

=  4  cos^  ct  —  3  cos  a. 
sin  3  a  =  3  cos^  «  sin  a  —  sin^  a 

=  3  sin  a  —  4  sin^  a. 

The  functions  of  4  «  and  of  higher  multiples  of  a  are  as  readily 
found.  The  simplicity  and  beauty  of  the  method  appears  on 
comparison  with  that  of  Art.  71. 

Examples 

1.  Show  that  cos«+^'s;"«  ^  cos  («-/?)  +  /  sin  («  -  S). 

cos  (3  +isin^ 

2.  Show  that  (  cos  "^  ^ 1-  i  sin     ^    —  )    =  cos  cc  +  i  sin  a. 

\  n  n      J 

3.  Show  that  f  cos  ^  ^^  "^  "  +  f  sin  -  ^'^  +  ^V  =  cos  a  +  «  sin  a,  where  /[:  is 

\  n  n        J 

any  integer. 

2  y^TT  4-  C£ 

4.  Show  that  the  angle  — — ~ —  has  n  different  values  as  k  takes  the  suc- 

n 
cessive  values,  0,  1,  2,  •  •  •  n  —  1  (n  being  a  positive  integer).      Show  also  that 
for  all  integral  values  of  k  outside  these  limits,  the  terminal  sides  of  the  angles 
coincide  with  those  of  the  n  angles  already  found. 

5.  Since  cos  0  -\-  i  sin  0  =  1,  find  the  ?i  different  nth  roots  of  1,  of  which  all 
but  one  are  imaginary.  Making  use  of  the  tables  of  natural  sines  and  cosines 
compute  for  n  =  2,  3,  4,  6. 

6.  Since  cos  tt  +  «  sin  tt  =  —  1,  find  the  n  different  nth  roots  of  —  1,  of 
which  all  but  one  are  imaginary  when  n  is  odd,  and  all  imaginary  when  n  is 
even.     Compute  for  n  =  2,  3,  4,  6. 

74.  Graphical  representation  of  complex  numbers.  An  interest- 
ing application  of  De  Moivre's  theorem  is  found  in  the  graphical 
representation  of  complex  numbers,  devised  by  Wessel,  a  Danisli 
mathematician,  and  published  by  Argand  in  1608.  The  treatment 
of  this  topic  belongs  rather  to  the  courses  in  algebra  and  function 
theory.  (See  Rietz  and  Crathorne's  "Algebra.")  Only  so  much 
of  the  rudiments  of  the  method  will  be  developed  here  as  possess  a 
trigonometric  interest. 

A  pure  imaginary  is  an  indicated  square  root  of  a  negative 
number.     A  complex  number  is  an  indicated  sum  of  a  real  number 


104  ANALYTIC   TRIGONOMETRY 

and  a  pure  imaginary.  All  pure  imaginaries  can  be  expressed  in 
the  form  «/^,  and  all  complex  numbers  in  the  form  x  +  yi.  Here 
^  =  V—  1,  so  that  z^  =  —  1 ;  while  x  and  y  are  real  numbers,  either 
rational  or  irrational. 

Argand's  method  makes  use  of  a  pair  of  mutually  perpendicu- 
lar axes.  The  Argand  diagram  must  not,  however,  be  confused 
with  the  Cartesian  scheme  of  coordinates. 

All  real  numbers,  rational  or  irrational,  are  represented  by  dis- 
tances from  the  origin  to  points  in  the  horizontal  axis,  called  now 
the  axes  of  reals,  positive  to  the  right,  negative  to  the  left.  To 
every  real  number  corresponds  a  point  in  this  axis,  and  conversely, 
to  every  point  in  this  axis  corresponds  a  real  number.  Thus  there 
is  said  to  be  a  one-to-one  correspondence  between  the  totality  of 
real  numbers  and  the  totality  of  points  in  the  line. 

All  pure  imaginaries  are  represented  by  distances  from  the 
origin  to  points  in  the  vertical  axis,  now  called  the  axis  of  imagi- 
naries, points  above  and  below  the  origin  giving,  respectively, 
positive  and  negative  coefficients  for  the  imaginary  unit  factor 
i  =  V—  1.  Here  again  there  exists  a  one-to-one  correspondence 
between  the  totality  of  pure  imaginaries  and  the  totality  of  points 
in  the  vertical  axis. 

Notice  that  the  origin  alone,  of  all  points  in  the  plane,  is  on 
both  axes.  The  number  zero  belongs  to  both  systems.  With  this 
single  exception,  no  pure  imaginary  can  equal  a  real  number,  since 
the  directions  of  the  two  axes  are  essentially  different. 

In  order  to  represent  the  complex  number  x  +  yi  recourse  must 
be  had  to  the  method  of  adding  coplanar  but  non-collinear  directed 
line  segments  employed  in  the  graphical  composition  and  resolution 
of  forces  in  physics.  Since  directed  line  segments  may  undergo 
translation,  the  segment  yi  may  be  placed  with  its  initial  point 
upon  the  terminus  of  the  segment  x.  The  complex  number  is 
therefore  represented  by  the  right  line  segment  (radius  vector)  v 
from  the  origin  to  the  resulting  terminus  of  the  segment  yi.  For 
y—^  we  have  real  numbers,  for  a:  =  0  we  have  pure  imaginaries. 

As  the  lengths  of  the  horizontal  segment  x  and  the  vertical 
segment  ^^  measure  respectively  the  magnitudes  of  the  reals  and  the 
pure  imaginaries,  so  the  length  of  the  radius  vector  v  may  be  said 
to  measure  the  absolute  magnitude  of  the  complex  number 
v  —  x-\- yi.  This  is  called  the  absolute  or  numerical  value  of  v, 
and  is  denoted  by  the  letter  r.  Evidently  all  points  on  the  unit 
circle  about  the  origin  possess  the  absolute  value  1. 


COMPLEX  NUMBERS 


105 


The  directed  line  segment,  or  radius  vector,  v  makes  in  general 
an  oblique  angle  with  the  axis  of  reals,  and  its  direction  is  deter- 
mined by  the  angle  it  forms  with  the  positive  axis  of  reals.  This 
angle  is  denoted  by  ^,  and  is  called  the  amplitude  of  the  complex 
number.  All  points  lying  on  the  same  radius  have  a  common 
amplitude,  while  radii  vectores  extending  from  the  origin  in 
opposite  directions  have  amplitudes  differing  by  tt.  All  positive 
real   numbers   have  the  amplitude  0 ;    negative  reals,   ir ;    pure 

TT  3  TT 

imagmaries,  —  or  -— -. 

The  right  triangle  formed  by  a;,  ^,  and  v  yields  the  relations 


6  =  arctan-1 

X 


\Y 


x  =  r  cos  ^,  3/  =  ^  sin  6, 

We  may  write  interchangeably, 

v^  ov  x  +  yi^  or  r  (cos  ^  +  ^  sin  ^). 

The  expression  cos  6  -\-i  sin  d  consequently  denotes  a  unit  segment 
(complex  unit)  with  the  amplitude  ^,  while  r  is  a  purely  arith- 
metical factor. 

Conjugate  complex  numbers,  x  -f  yi  and  x  —  y%  evidently  have 
the  same  absolute  value  and  amplitudes  which  are  negatives  of 
each  other. 

Addition  is  effected  graphically  by  placing  the  initial  point  of 
the  second  segment  upon  the 
terminus  of  the  first  and  con- 
necting the  initial  point  of  the 
first  to  the  terminus  of  the 
second.     Thus  in  Fig.  70, 

=  ^1  +  %i  +  ^2  +  % 

=  (^1  +  ^2) +^'(^1  +  ^2)- 
The  values  of  r  and  d  in  terms  of  /-j,  r^^  6^  and  0^  are  readily  deter- 
mined, but  exhibit  little  of  present  interest.     Suffice  it  to  point 

out  that 

r  <  rj  +  9-2, 

e^e^  +  e^. 

Subtraction  reduces  at  once  to  addition  on  reversing  the  sub- 
trahend segment. 


Fig.  70. 


106  ANALYTIC    TRIGONOMETRY 

On  attacking  the  problem  of  multiplication,  we  must  define 
the  product  of  a  directed  rectilinear  segment  by  the  imaginary 
unit  {  as  a  segment  of  equal  length  turned  through  a  positive 
right  angle.  Thus  v  =  x-{-iy  =r  (cos  6  -\-i  sin  ^)  multiplied  by  i 
gives 

=  —  y  -\-ix=r  cos  (  ^  +  ^  )  +  2  sin  (  ^  +  ^  j  • 


2        7  V2 


The  absolute  value  is  unchanged,  while  the  amplitude  is  in- 


ir 


creased  by  —  •     This  is  consistent  with  the  original  scheme  of  rep- 

A 

resentation,  since  reals  multiplied  by  i  give  pure  imaginaries,  and 
these  multiplied  by  i  give  —  1  times  the  original,  i.e.  the  original 
radius  vector  reversed. 

Multiplying  a  directed  segment  by  a  positive  real  number 
simply  stretches  it,  multiplying  its  length  and  leaving  its  direc- 
tion unchanged.     Multiplying 

V  =x-\-iy  =  r  (cos  0  -\-i sin  ^)  by  k^  we  get 

v'  =  hv  =  kx  -\-  iky  =  kr  (cos  0  -\-i  sin  ^). 

The  absolute  value  is  multiplied  by  the  factor  k,  while  the  ampli- 
tude is  unchanged. 

Multiplication  of  one  complex  number  by  another  is  effected 
by  combining  the  two  processes  just  described,  applying  the  asso- 
ciative and  distributive  laws.     Thus 

v  =  v^'V^=  (^1  H-  iyi)  '  (a^2  +  ^^2) 

=  ^1  (^2  +  ^>2)  +  *>i  (^2  +  %) 

=  (.V2  -  ViVi)  +  ^^1^2  +  ^2^1) • 

Using  the  other  notation  and  applying  De  Moivre's  theorem, 

v  =  v^  •  V2  =  r^  (cos  ^j  +  i  sin  ^j)  •  r^  (cos  0^  +  i  sin  0^^ 

=  r^r^  '  [cos ((9i  +  0^-)  +  i  sin ((9^  -hO^)-]. 

Figure  71  illustrates  the  multiplication  of  5  —  2  ^  by  2  4-  3  ^. 
The  product  is  shown  to  be  16  +  11  i.  We  have  then  the  law  that 
the  absolute  value  of  the  product  of  two  complex  numbers  equals 
the  product  of  their  absolute  values,  while  the  amplitude  of  the 
product  equals  the  sum  of  their  amplitudes. 


COMPLEX  NUMBERS 


107 


The  inverse  process  of  division  is  readily  performed,  with  the 
result 


or 


V  =  -1  = 

^2 


X,  +  ly^  _  x^x^^  +  y^y^       .  x^y^  -  x^y^ 


^2  +  ^Vl  ^2    +  ^2 

r^(cos  0^  +  ^  sin  Q^ 
r^  (cos  ^2  +  ^  sin  ^2) ' 


+  ^ 


+  ^2^ 


i;  =  ^  [cos  (6'i  -  6>2)  +  ^  sin  ((9i  -  6>2)]  • 
^2 
The  absolute  value  of  the  quotient  is  equal  to  the  quotient  of 
the  absolute  values,  while  the  amplitude  of  the  quotient  is  equal 
to  the  difference  of  the  amplitudes. 
We  have  further, 

vz=v{'  =  r-^  (cos  nO^  +  i  sin  nO^. 

The  absolute  value  of 
the  power  is  equal  to 
the  power  of  the  abso- 
lute value,  while  the 
amplitude  of  the  power 
is  equal  to  the  ampli- 
tude of  the  number 
multiplied  by  the  index 
of  the  power.  Here 
"  power  "  is  used  to  de- 
note the  result  of  affect- 
ing the  number  by  the 
exponent  n^  whatever 
the  value  of  n.  This 
includes  both  involu- 
tion and  evolution.  In 
particular  let  n  be  the 
reciprocal  of  a  positive  integer  m. 


Fig.  71. 


V-  m/ 


Then 

cos—i  + 
m 


mj 


But  Vj  is  just  as  well  and  exactly  represented  by 

r  [cos  (2  ^TT  +  ^)  +  i  sin  (2  kir  +  ^)], 

where  h  is  any  integer.     Thus  the  mth  root  just  found  is  only  one 
of  an  infinite  number,  all  given  by  the  form 


m/-r        2kir  +  0. 
Vr.    cos =^— i 


I 


+  i  sin 


^kir 


m 


±6,1 

^  J 


108  ANALYTIC    TRIGONOMETRY 

in  which  k  assumes  all  integral  values.  This  form  gives  m  dif- 
ferent values  for  the  root,  corresponding  to  A;  =  0,  1,  2,  •••  m  — 1. 
All  the  others  are  repetitions  of  these  m  roots,  since  the  terminal 
sides  of  all  the  other  amplitude  angles  will  coincide  with  the  ter- 
minal sides  of  the  m  amplitudes  specified. 

Hence  every  complex  number  has  m  different  mth  roots,  whose 
common  absolute  value  is  the  arithmetical  mth  root  of  the  absolute 
value  of  the  number,  while  their  amplitudes  have  the  m  different 
values, 

e.     ^TT  +  e^     4  7r  +  6>^  2fm-l)7r  +  (9^ 

m         m  m  m 

all  less  than  2  tt. 

In  the  special  case  of  any  positive  real  number  a^j,  whose  am- 
plitude is  therefore  zero,  we  obtain  m  different  mth  roots  with  the 
common  absolute  value  Vr^,  which  is  called  the  principal  value  of 
Va^j,  and  the  m  different  amplitudes, 

/^    2  TT     4  7r      Gtt         2(m  — l)7r 
u,        ,         ,         ,  •••  • 

m        m        m  m 

Only  one  of  these  is  real,  the  first,  and  it  is  called  the  principal 
mth  root  of  the  positive  real  number. 

The  student  should  construct  figures  to  illustrate  the  foregoing 
theorems.  Still  another  analytic  notation  for  complex  numbers 
will  be  brought  out  in  Art.  75. 

Examples 

1.  Represent  by  Argand's  diagrams^the  numbers  2,  —3,  3i,  —  4i,  3  +  5t, 

4  -  3 1,   -  2  +  i,   -  5  -  3  i,  4  +  VZ^  Vs  -  VIT?. 

2.  Write  the  numbers  the  termini  of  whose  radii  vectores  have  the  Carte- 
sian coordinates  (3,4),  (-3,2)^  (7,-3),  (-5,-2),  (6,0),  (0,5), 
(-2,0),    (0,  -6),    (0,0),   (V^,    V5). 

3.  Find  the  absolute  values  and  the  amplitudes  (expressed  in  degrees  and 
minutes)  of  the  numbers  in  examples  1  and  2. 

4.  Describe  the  situation  of  the  number  points  which  have  :  (1)  the 
common  absolute  value  3 ;  (2)  the  common  amplitude  30° ;  (3)  the  amplitudes 
45°  and  225°. 

5.  Perform  graphically:  (3  +  40  +  (7  -  2  0  ;  (-3  +  2/)  +  (6  -  3  {)  ; 
(7  -  3  0  -  (4  +  2  i)  ;  (3  -  2  0  -  (-  6  -  3  i)  ;  (5  +  2  i)  +  (3  -  4  i)  -  (6  -  3  {). 

6.  Perform  graphically,  taking  the  first  factor  in  each  case  as  the  multi- 
plier:  3.(5  +  2  0;  I -(3 +  50;  2  t- (6-30;  -4.(2+50;  -6  i .  (3+ 2  0; 
(4+  2  0  •  (3  +  4  0  ;    (3  +  4  0  -(4  +  20. 


EULER'S  EXPONENTIAL  VALUES  109 

21  +  ;      6  -  17 1 


7.   Construct  the  quotient  of 


3  +  2f       4-3t 


8.  Construct:    (3  +  202;     l-l^i^V-,    (l-iV'dy;i\ 

9.  Find  by  construction:    V7-24i;     \/-  119  +  120  i;    (-5  +  12i)^' 

^/ITl;     ^;     ^16. 

10,  Write  the  general  solution  of  the  binomial  equation  :  x"  —  a"  =  0. 

11.  Find  all  the  roots  of  the  equations  a:^  —  1  =  0 ;  x'^  -{-  1  —  0;  x^  —  1  =  0 ; 
a;3  -  8  =  0. 

75.  Exponential  values  of  the  trigonometric  functions.  The 
first  form  of  De  Moivre's  theorem,  Art.  73,  Eq.  (1),  may  be  written 
symbolically, 

which  is  read,  function  of  a  times  (the  same)  function  of  /9  equals 
(the  same)  function  of  (a  +  /3)  ;  or,  the  product  of  the  (same) 
functions  of  two  numbers  equals  the  (same)  function  of  the  sum 
of  the  two  nuifibers.  Now  this  is  identically  the  characteristic 
relation  or  law  governing  the  exponential  function,  that  is,  a 
function  of  the  form  a^ ;  thus. 

For  reasons  discussed  in  Art.  77,  it  is  found  that  instead  of  the 
more  general  function  a%  we  must  place 

cos  a-\-i  sin  a  =  e'%  (1) 

where  e  =  2.71828183  •••  is  the  base  of  the  Naperian,  or  natural, 
system  of  logarithms  given  in  Art.  23. 

Note  that  the  law  of  exponents,  derived  for  positive  integral 
exponents,  and  assumed  to  hold  also  for  negative,  fractional,  and 
irrational  exponents,  is  still  further  assumed  for  exponents  which 
are  pure  imaginaries  and  complex  numbers.  As  in  the  former 
cases,  the  significance  must  be  determined  in  conformity  to  the 
action  of  the  assumed  law.     Indeed,  the  law  defines  the  function. 

Since  cos  a  —  i  sin  a  = ;— : ,  we  have  also 

cos  a  +  ^  sin  a 

cos  a—i  sin  a  =  er^°-.  (2) 

Adding  and  dividing  by  2,  we  obtain 

cosa  = ;  Ko) 


110  ANALYTIC   TRIGONOMETRY 

again,  subtracting  and  dividing  by  2  ^^ 

sm  a  =  — ^j—  .  (4) 

These  values  were  first  given  by  Euler  in  1743.  Starting  from 
these  two  exponential  values  as  fundamental  definitions,  and  de- 
fining further 

,              sin  rt         ,             1                         1  1 

tana  = ,  cota  = ,  seca  = ,    csca  = 


cos  a  tan  a  cos  a  sm  a 

it  is  possible  to  develop  all  the  laws  and  formulas  of  trigonometry 
as  contained  in  Arts.  59  and  63-71,  quite  apart  from  any  geo- 
metric meaning  attached  to  the  functions  or  their  argument  a. 
The  analogous  derivation  of  those  trignometric  theorems  de- 
pendent on  the  periodicity  of  the  trigonometric  functions  involves 
the  periodicity  of  the  logarithm,  and  is  therefore  postponed  until 
the  later  mathematical  study  of  the  student. 

A  third  notation  for  complex  numbers  now  becomes  manifest.; 
for 

v  =  x  +  iy  =  r  (cos  6  -\-  i  sin  ^)  =  re^^. 

The  consequent  theorems  regarding  the  absolute  values  and  am- 
plitudes of  products,  quotients,  powers,  and  roots  follow  readily, 
and  should  be  worked  out  by  the  student. 

Examples 

1.  Find  the  exponential  values  of  tan  a,  cot  a,  sec  a,  esc  a. 

2.  Derive  from  the  exponential  values  the  laws  sin^  a  +  cos^  a  =  1,  etc.,  of 
Art.  59. 

3.  Derive  from  the  exponential  values  the  formulas  of  Arts.  63-71. 

4.  Derive  from  the  exponential  notation  the  laws  for  the  absolute  values 
and  amplitudes  of  products,  quotients,  powers,  and  roots  of  complex  numbers. 

76.    Hyperbolic  functions.    Closely  allied  to  Euler's  forms  of  the 
last  article  are  the  two  interesting  and  important  forms. 


2  2 

They  are  called,  by  analogy,  the  hyperbolic  cosine  and  hyperbolic 
sine.     Thus,  employing  the  customary  notation, 

cosh  a  = ,   sinha= — 


HYPERBOLIC   FUNCTIONS  111 

The  remaining  hyperbolic  functions  are  defined  from  these,  just  as 
in  Art.  75  : 

tanh  a  =  ^IBIL^,  coth  a  =  —1—,  sech  a  = -—,  csch  a  =  ___. 

cosh  a  tanli  a  cosh  a  sinh  a 

A  very  simple  relation  exists  between  the  hyperbolic  and  the 
circular  (^i.e.  ordinary  trigonometric)  functions.     Evidently 

cosh  a  =  cos  ^a, 

sinh  a  =  —  i  sin  m, 

tanh  a  =  —  ^  tan  ia ; 
and  conversely, 

cos  a  —  cosh  za, 

sin  a  =  —  i  sinh  m, 

tan  a  =  —  {  tanh  za. 

To  each  formula  of  Chapter  YIII  corresponds  a  formula  for  the 
hyperbolic  functions,  which  may  be  deduced  either  directly  from 
the  exponential  definitions,  or  by  substituting  the  values  just 
given  in  the  formulas  for  the  circular  functions.  The  student 
should  derive  these  formulas  by  both  methods. 

The  analogue  to  De  Moivre's  theorem  is 

(cosh  a  -h  sinh  a)"  =  cosh  na  -\-  sinh  na. 

Cosh  a  and  sinli  a  possess  an  imaginary  period  2  7^^,  since 
gM  _  ^u+2kni^  jf.  being  any  integer.  (See  treatises  on  the  theory  of 
functions.) 

77.  Exponential  and  trigonometric  series.  In  the  present 
article  values  in  the  form  of  infinite  series  will  be  derived  for 
certain  exponential,  logarithmic,  and  trigonometric  functions. 
In  the  proof,  however,  the  use  of  the  binomial  formula  and  the 
manipulation  of  the  series  introduce  a  lack  of  rigor  requiring  ex- 
tended consideration  in  the  subsequent  courses  in  algebra,  the 
calculus,  and  the  theory  of  functions. 

(1)  Exponential  series. 

Expanding  by  the  binomial  formula,* 

n)  ~         ll'^^        2!         ^  3!  '  n^ 

*The  symbol  \k,  or  k !,  is  used  to  denote  the  product  1  •  2  •  3  ••■  A:,  where  k  is 
any  positive  integer,  and  is  read  "  factorial  A;". 


112 


ANALYTIC   TRIGONOMETRY 

.2 


(2) 


Now  as  n  becomes  infinite,  the  binomial  factors  [  1 ),  f  1  —  — ), 

etc.,  all  approach  the  common  limit  1,  and  we  shall  have,  in  the 
limit, 

,-  =  limrA+^Y1=l  +  ^  +  ^  +  |^+....  (1) 

n=^[\       nJ  J  1!      2!      3! 

This  series  is  convergent  for  all  finite  values  of  x.    (See  Rietz  and 
Crathorne's  "Algebra.") 
For  a;  =  1  we  get 

1,1,1.1,1, 

^=^  +  l!  +  2!-^3!  +  T!-^- 

The  terms  diminish  rapidly  in  value  and,  when  expressed  deci- 
mally, the  value  of  e  is  found  to  be  2.71828183  •••. 

The  series  for  e^  is  valid  also  for  negative  and  imaginary  values 
of  X  ;  thus,  substituting  successively  —x,  ix,  and  —ix  for  a;,  we  have 

*     =^-T!  +  2!-8T+-' 

/y>^  /"Y*^  /^O  /y%  /y»t>  /y*0  /y»7  I 

^    =^-2!  +  4!-6!  +  -  +  ln-^  +  6!-f!+-} 

_,>      -,       x^  ,   x^      x^  ,  .r  X       a^  ,  x^      x'^  ,        ~] 

^     ='-2!  +  4!-0l+--iT!-^  +  5!-7!+-J- 

(2)  Logarithmic  series. 

From  the  expansion  just  obtained  for  e"^  can  be  derived  a  series 
for  log,  (1  +  ^). 

Since*  ^t^  =  e^l«ge«, 

we  have      ^^  =  1  +  ^  (log, ^)  +  |y  (loge ^)^  +  ^y  (log,  uy  +  -- -. 
Placing        u=l  +  t/f 

(l+y)-=l  +  ^log,(l+y)  +  ^[log,(l  +  y)P 

+  ^Doge(l+y)?+-- 

*  Let  w  =  u*.     Taking  logarithms  to  base  e,  we  have  loge  w  =  x  log«  u.    Now 
taking  exponentials  to  base  e,  lo  =  w*  =  e*  log*". 


TRIGONOMETRIC    SERIES  113 

Expanding  the  first  member  by  the  binomial  formula, 

(1+^)^  =  1  +  — y+    ^  ^^     ^f  +  -^ ^y^ -f+  •••• 

Picking  out  and  equating  the  coefficients  of  x  in  the  two  expres- 
sions, the  required  expansion  is  obtained, 

log«(l+2/)  =  f-f +  |-J+-.  (8) 

This  series  is  convergent  for  —  1  <  ?/  ^  +  1. 

If  w  =  loga u^  we  have  a"'  =  u;   wlience,  taking  logarithms  to 

base  g,  w  log^  a  =  logg  u.     Therefore  w  =  log«  u  = •  log.  u, 

log,  a 

Substituting  from  (3)  we  see  that 

(3)   Trigonometric  series. 

From  De  Moivre's  theorem 

cos  mO  -\-i  sin  mO  =  (cos  6  -\-i  sin  ^)"*. 
Expanding  b}^  the  binomial  formula  and-  separating  the  real  terms 
from  the  imaginary, 

cos  mO  +  i  sin  md  =  cos"^  6  -  ^^^  ~  ^^  cos'^-s  ^  sin2  0 

.  w(m— l)(m  —  2)(m— 3)        m-A  a    -   4.  a 
+  — ^^ — ^ cos"^  ^  a  sin^ a  •" 

4! 

-f  ^ f  — -  cos"*  1  6  sm  6  —  — ^ ^ ^  cos"*-3  0  gm^  0  j^  ...  \ 

Equating  separately  the  real  and  the  imaginary  parts, 
cos  mO  =  cos"*  0  -  ^^^"-^^  cos"*-^  0  sin2  (9  4-  •••, 

sinmu==—j  cos"*  ^^sm^ ^^ /y^ ^cos"*  ^osin^6-{-  •••. 

Place  now 

m$  =  a,  so  that  ^  =  —  ; 

co.„  =  cos"Q-^(^cos-Qsi,.Q+..., 


sm 


1 !  \wy        Vm/ 


m(m-l)(m-r2) 
3! 


-<^)^''<3^-- 


114  ANALYTIC   TRIGONOMETRY 

Now  let  0  approach  the  limit  zero  and  m  become  infinite,  while 
still  obeying  the  condition  that  mO  =  a,  where  a  remains  finite. 

By  Art.   12,  cos—,  cos^— ,  etc.,  approach  the  limit  1  as  m  becomes 

mm  .    . 

infinite,   and  in    the    calculus    the    same    is  shown  for  cos"*f  — j, 
cos^^-if-Y  etc. 


Again,  m  sm    —   =  •, 

\mj  u 

■      m(m-l)sin2('"'\=a(«-6>)./'^Y, 
m{rn  -  1)  (m  -  2)  sin^/"-^  =  «(«  -  ^)(«  -  2  l9)  •  f^}^\\  etc. 

Since  li^^^  ^ =  1,  the  limits  approached  by  these  expressions,  as 

^  =  0,  are  a,  a\  a^,  etc. 

Making  these  substitutions,  we  obtain,  in  the  limit, 
^        «2        «4        ^6  . 

cos«  =  l--  +  jj-^j+...,  (6) 

'^""  =  r!-3!+6T-^+--  <^^> 

These  series  are  convergent  for  all  values  of  a. 
Tan  a  may  also  be  expanded  into  the  series 

*'^""  =  i  +  F  +  i5  +  W+--  ('> 

It  will  be  noticed  that  the  series  for  cos  a  contains  only  even 
powers  of  a,  while  those  for  sin  a  and  tan  a  contain  only  the  odd 
powers  of  a.      (See  the  third  example  worked  out  in  Art.  62.) 

The  assumption  of  Art.  75  may  now  be  justified.  For,  on  sub- 
stituting for  e'^,  e"'^  sin  a:,  and  cos  a;  their  expansions  in  series,  we 
obtain 

cos  x-\-i  sin  x  =  e'^, 

cos  ic  —  ^  sin  a;  =  e~"^, 
and  cos  x  — 


^rx 

4-  e-'' 

2 

^ix 

-  e-'^ 

sma: 

2i 


COMPUTATION   OF   TABLES     •  115 

(4)  Hyperbolic  series. 

From  the  analogous  relations  the  expansions  for  the  hyperbolic 
functions  are  readily  obtained. 

ri^i  1  e'^  +  e~'^      1    ,    ^2      ^4       ^6 

Ihus  cosha  =  -^^=l  +  -  +  -  +  -+  .-,  (8) 

Examples 

1.  By  substituting  —  a  for  a,  find  the  series  for  sin(—  a),  cos(—  a), 
tan  (—  a),  and  by  comparison,  verify  the  corresponding  relations  of  Art.  62. 

2.  By  substituting  ia  for  a,  verify  the  relations  of  Art.  76,  cosh  a  —  cos  la, 
etc. 

3.  Using  two  terms  of  the  expansions  for  sin  a  and  cos«,  and  retaining 
only  powers  of  a  below  the  fifth,  obtain  an  approximate  verification  of  the  fol- 
lowing formulas : 

sin^a  +  cos^a  =  1,  sin  (a  +  /?)  =  •••,  cos  (a  +  y8)=  •••,  sin  2  a=  ••-,  cos  2  a  =  •••• 

4.  Do  as  required  in  example  3  for  the  hyperbolic  functions. 

5.  Repeat  examples  3  and  4,  using  three  terms  of  the  series  and  retaining 
powers  of  a  below  the  seventh,  thus  arriving  at  a  closer  approximation. 

78.  Computation  of  trigonometric  tables.  The  numerical  values 
of  the  sine,  cosine,  and  otlier  trigonometric  functions  of  angles 
from  0°  to  90°,  as  tabulated  in  Table  III,  may  be  calculated  by 
means  of  various  trigonometric  formulas,  or  better,  by  the  use  of 
the  series  derived  in  Art.  77. 

Euler  gave  the  following  series,  carrying  the  computation  to 
28  decimal  places  and  a  corresponding  number  of  terms  :   Place 

a  =  m  '  -^  in  the  series  for  sin  a  and  cos  a ;  whence 

sin  ^w.  1^  =  1.570  796m  -0.615  964^3 

+  0.079  693  m^  -  0.004  682  m'^ 
-f-  0.000  160 m^- 0.000  004m" 

+ , 

cos  C^  •  I)  =  1-000  000   -  1.233  700  m^ 
+  0.253  699  m^  -  0.020  863  m^ 
-{-  0.000  919  m^  -  0.000  025  m^^ 
+ 


116  *     ANALYTIC   TRIGONOMETRY 

We  need  to  calculate  the  sines  and  cosines  of  angles  up  to 
45°  only,  so  that  m  is  a  fraction  and  always  less  than  |-.  The 
terms  of  the  series  converge  rapidly  and  a  few  terms  suffice  to  give 
values  correct  to  a  small  number  of  decimal  places. 

More  extended  discussion  of  this  topic  may  be  found  in  Hob- 
son's  "Trigonometry,"  Chap.  IX;  Todhunter's  "Plane  Trigonome- 
try," Chap.  X ;   and  other  advanced  treatises  on  trigonometry. 

The  series  for  log  (1  +  ^)  converges  too  slowly  for  convenient 
calculation,  but  a  modified  form  is  easily  obtained.  ^  Manifestly 

/2 

~  8 


log  (1  -  ^)  =  -  ^  -  |-     ^ 


and 


log  l^f  =  1^'g  (!  +  «/)-  log  (1  -  ^) 


—  9 


I  +  -8+5  + 


} 


Place  y  = ,  Avhence         ^  =  ;  then 

2i;  +  l  1  —  y         ^ 

log  !i±i  =  2^-1-+ 1 + 1 :  +  ...), 


or 


log  (v  +  1)  =  lo^  V  -h  2r — - —  H h — +  •••T 

This  series  converges  rapidly  and  by  it  log  2  can  be  computed 
from  log  1  =  0,  log  3  from  log  2,  etc.  Logarithms  of  composite 
numbers  can  be  checked  by  adding  the  logarithms  of  their  factors. 

79.  Proportional  parts.  In  using  the  logarithmic  and  trigo- 
nometric tables  it  Avas  assumed,  as  stated  in  Art.  2^,  that  for 
small  differences  in  the  number,  the  differences  in  the  logarithm 
are  proportional  to  the  differences  in  the  number,  and  that  like- 
wise, for  small  differences  in  the  angle,  the  differences  in  the  sine 
(or  other  trigonometric  function,  or  logarithmic  function)  are  pro- 
portional to  the  differences  in  the  angle. 

We  have 

log  (a;  +  8)  -  log  a;  =  log^±^  =  log  f  1  + -^ 

X  '    \        xj 

^S_j5^      J3 ^ 

X  2X^  SX^         4:X^ 

__  8  (Approximately  for  small 

X  values  of  3.) 


PROPORTIONAL   PARTS.     INVERSE   FUNCTIONS  117 

Therefore,  we  have  approximately 

log  ix-\-8^)-lo^x^  S^  /^  =  ^1 ,  (13 

log  (x  +  82)  -  log  X      hjx      S2' 

for  small  differences. 
Again, 

sin  (3  -}-  |9)  -  sin  (9  =  2  cos  [o  +  -")  sin  | 

=  cos  Q  •  3.     (Approximately  for  small 

values  of  3.) 
Hence,  approximately 

sin  (0  4-  3j)  —  sin  0      \  cos  ^      \ 


sin  (^  +  ^2)  —  sin  Q      h^  cos  d      h^ 


(2) 


For  the  other  functions,  the  proof  follows  exactly  similar  lines, 
and  can  easily  be  supplied  by  the  student. 

Full  discussion  along  this  line  may  be  found  in  Loney's  "  Plane 
Trigonometry,"  Chap.  XXX ;  Lock's  "  Higher  Trigonometry," 
Chap.  VIII ;  Hobson's  "  Trigonometry,"  Chap.  IX  ;   etc. 

80.  General  inverse  functions.  In  Art.  14  only  acute  angles 
were  under  consideration,  so  that  the  relations 

m  =  sin  a,  a  =  arcsin  m, 

expressed   a  one-to-one  correspondence.     In  other  words,  under 
the  condition  that 

0°<«<90°,  05m<l, 

to  each  value  of  «  there  corresponds  one  and  only  one  value  of 
m  and  conversely. 

On  considering  the  general  angle,  it  became  evident  that  to 
any  one  angle  there  corresponds  one  and  only  one  value  of  the 
sine  (or  other  function),  but  that  to  one  value  of  the  sine  (or 
other  function)  correspond  many  angles.  We  define  arcsin  m, 
arccos  m,  arctan  ?w,  etc.,  as  the  numerically  smallest  angle  having 
the  given  sine,  cosine,  tangent,  etc.     It   follows  that  arcsin  m^ 

arctan  w,    arccot  m,   arccsc  m   always   lie    between   —  —  and  +— , 

while   arccos  m  and   arcsec  m    always   lie   between    0  and    +  tt. 
These  are  called  the  principal  values  of  the  general  inverse  func- 


118  ANALYTIC    TRIGONOMETRY 

tions  Arcsin  w,  Arccos  w,  etc.     As  results  of   Arts.    61,  62,  we 
may  write,  if  k  is  any  integer, 

Arcsin  m  =  2  A^tt  +  arcsin  m, 

Arccos  m  =  2  kir  -{-  arccos  m, 

Arctan  m  =      Jctt  -{-  arctan  w, 

Arccot  m  —     Jctt  -\-  arccot  m, 

Arcsec  m  =  2  kir  +  arcsec  m, 

Arccsc  m  =  2k7r  -{-  arccsc  m. 

Similar  relations  exist  for  the  inverse  hyperbolic  functions,  the 
periods  being  imaginary,  2  kiri  and  kiri. 

From  the  relations  of  Art.  76  may  be  derived  the  following : 

arccos  m  =  (  ±  )  ^  inv  cosh  m^ 

arcsin  m  =  —  i  inv  sinli  im^ 

arctan  m  =  —  z  inv  tanh  iyn  ; 
and 

inv  cosh  m  =  (  ±  )  ^  arccos  w, 

inv  sinh  m  =  —  i  arcsin  im^ 

inv  tanh  m  =  —  i  arctan  im. 

81.  Logarithmic  values  of  inverse  functions.  Since  the  circular 
and  hyperbolic  functions  are  expressible  as  exponential  functions, 
it  would  seem  that  the  inverse  functions  should  be  expressible  as 
logarithmic  functions.  Such  is,  indeed,  the  case,  and  the  desired 
values  may  be  found  by  solving  for  a  the  forms  given  in  Arts.  75 
and  76. 

( 1 )    Circular  functions . 

If,  for  example, 

z  —  Sin  a  = -— — , 

2  I 

we  have  the  quadratic  equation  in  e'% 

g2.«  _  2  {ze^<^  -1  =  0, 

whose  roots  are  

e'"  =  iz  ±  Vl  —  z^. 

Choosing  the  upper  sign,  and  taking  logarithms  to  base  e,  we 

^^^  ia  =  log  {iz  +  vr^^), 

whence  ^  ^  ^^^^.^  ^  _  _  .  ^^^  ^  .^  _^  ^^^^  ) 

gives  the  principal  value  of  the  arcsine. 


INVERSE   HYPERBOLIC   FUNCTIONS  119 

(2)  Hyperbolic  functions. 

As  may  be  expected,   the  values  of   the  inverse   hyperbolic 
functions  are  real  in  form.     Thus  from 

z  =  smh  cf  = , 


we  get,  on  solving, 

a  —  inv  sinh  z  =  log  {z  +  Vs^  +  1  )• 

Examples 
Obtain 


1.  arccos  z  =  —  i  log  {z  +  Vz^  —  1  )• 

2.  arc  tan  s  =  -  i  log  ^  +  ^^ 


2        "  1  -  I. 


iz 


3.  inv  cosh  z  —  log  2  {z  +  va;^  —  1  ). 

1  1-4-2 

4.  invtanhs  =  -  log  =--^. 

2  ^  \-z 


REVIEW   EXERCISES 

1.  To  what  quadrant  do  the  following  angles  belong :    560°,   653°,   1030°, 
425°,    -1260°? 

2.  To  what  quadrant  do  the  following  angles  belong:  — ^,   r!L_^  8^, 

5  o 


13  TT^' 

3    ' 

25^,    ■ 

4    '     - 

3 

3. 

Reduce  to  radians 

:   75° 

300°,   -250°,   2000°,   465°  20'. 

4.  Reduce  to  the  degree  system  :   4^,   -  6^,    i^,    ^,     _  I^. 

o  o  2 

5.  Find  the  lengths  of  the  arcs  subtended  by  the  following  angles  at  the 

center  of  a  circle  of  radius  6  :  45°,  120°,  270°,  ^^,    ^^,    5^- 

'4  8  3. 

6.  A  polygon  of  n  sides  is  inscribed  in  a  circle  of  radius  r.  Find  the 
length  of  the  arc  subtended  by  one  side.  Compute  the  numerical  values  if 
r  =  10  and  w  =  3,  4,  5,  6,  8. 

7.  Taking  the  radius  of  the  earth  to  be  4000  miles,  find  the  difference  in 
latitude  of  two  points  on  the  same  meridian  300  miles  apart. 

8.  Find  the  difference  in  longitude  of  two  points  on  the  equator  1200 
miles  apart. 

9.  Find  the  distance  in  degrees  between  two  points,  one  of  which  is  800 
miles  due  north  of  the  other. 

10.  A  city  is  surrounded  by  a  circular  belt  line  5  miles  in  radius.  How 
long  will  a  train  require  to  go  at  a  speed  of  20  miles  an  hour  from  a  station  due 
east  of  the  center  to  one  due  northwest,  if  the  motion  is  clockwise ;  if  counter- 
clockwise ? 

11.  Find  with  the  protractor  the  angles  formed  successively  by  the  radii 
vectoresof  the  points  (3,  0),  (2,  4),  (-3,  5),  (0,  6),  (-4,  2),  (-  2,  1),  (5,  -3), 
(8,  0). 

12.  Find  with  the  protractor  the  angles  of  the  triangles  formed  by  the 
abscissa,  ordinate,  and  radius  vector  of  each  of  the  following  points  :  (4,  4), 
(1,3),  (3,  -3),  (-2,2),  (-4,  -8). 

13.  Find  by  measurement  the  coordinates  of  the  point  whose  radius  vector 
is  4  and  makes  an  angle  of  30°  with  the  positive  x-axis ;  5  and  120° ;  8  and  225°. 

14.  Find  by  measurement  the  length  and  inclination  angle  of  the  radii 
vectores  of  the  points  whose  coordinates  are  (2,  5),  (—5,  12),  (—  8,  —  15). 

120 


REVIEW  EXERCISES 


121 


15.  If  the  earth  were  assumed  to  be  a  plane,  and  one  degree  of  latitude  or 
longitude  were  60  miles,  what  would  be  the  distance  and  direction  from  a  point 
in  20°  N.  lat.,  60°  E.  long.,  to  one  in  50°  N.  lat.,  30°  E.  long. ;  from  a  point  in 
30°  S.  lat.,  15°  E.  long.,  to  one  in  45°  N.  lat.,  40°  W.  long.  ? 

Note.  This  assumption  is  made  by  navigators  as  a  basis  for  what  is  known  as 
Plane  Sailing.  In  Great  Circle  Sailing  the  earth  is  considered  a  sphere.  Let  the 
student  devise  a  system  of  coordinates  for  the  latter. 

16.  Find  by  measurement  the  six  trigonometric  functions  of  36°,  155°, 
285°,  -  130°. 

17.  Find  by  measurement  the  following  angles:  arccot  |  and  of  1st  quad- 
rant ;  arcsin  f  and  of  2d  quadrant ;  arccos  (  —  0.3)  and  of  3d  quadrant ;  arcsec 
0.6  and  of  4th  quadrant. 

18.  Find  the  lacking  functions  in  the  following  table : 


Angle 

Sink 

Cosine 

Tangent 

Cotangent 

Secant 

Cosecant 

Quadrant 

a 

-tV 

III 

ft 

f 

IV 

y 

-i 

ir 

a 

Y- 

III 

e 

_  2 

TIT 

<t> 

3 

T 

19.  Find  the  value   of       ^^^ "      +      ^"^  ^      ii  a  =  arcsin  -  and  of  2d 
quadrant.  1  -  tan «       l-cot«  5 

20.  Find  the  value  of  tan  /?  +  sec  y8  -  1    if  ^  ^  arccsc  f  -  — )  and  of   3d 
quadrant.  tan  ^  -  sec /?  +  1        ^  V      6  J 

21.  Find  the  value  of  tan^g  -  sin^ct   .^  ^  ^  arccot  f  -  I]  and  of  4th  quad- 
rant. '^^'^  ^     2; 

22.  Find  the  value  of  1  + cosy -2  secy  -^         arctan-^  and  of  1st  quad- 
rant. 3  + cosy +  2  secy  40 

23.  Express  ^^^^  "  +  ^^^^  ^  in  terms  of  tan  a. 

sec2  a  +  csc2  a 

24.  Express  4-sin^-3csc^  j^  ^^^^^  ^^  ^^^^^ 

sin  /8  -  1  -  6  CSC  /^  ^ 

25.  Express  Q  -  cos  «)  (1 -f  sec  «)  -^  ^^^^^  ^^  ^^^  ^ 

(1  -sin  a)(l  -fcsca) 

26.  In  the  following  identity  transform  the  first  member  into  the  second, 

(1  +  tan  y)  (cosy- cot  y)  ^  _  ^^^ 
(1  +  cot  y)  (sin  y  —  tan  y) 


122  REVIEW   EXERCISES 

27.  Show  that  (l-tana)(l  +  cot(x)  ^  _  ^^ 

(1  +  tan  a)(l  -  cotw) 

28.  Show  that   sin^  /g  +  cos^  /S      siii«  ^  -  cos^  ^  ^  _  ^  ^ .^^,  ^ ^^^,  ^^ 

Solve  the  following  equations  and  find  the  angle  in  degrees  : 

29.  4sin2y- tan^ynzO. 

30.  2  tan2  a  -  sec  a  =  4:. 

31.  4  CSC  ;8  +  cot2  y8  =  5. 

32.  tan2y  +  3cot2y  =  4. 

33.  sin2  a  +  sin^  /3  =  I,  cos"^  a  +  cos^  (3  =  0. 

34.  2  cos2  a  +  sin2  y8  =  2,  sin  a  +  cos^  y8  =  0. 

35.  For  what  range  of  values  of  a  between  0  and  2  tt  is  sin  a  +  cos  a  posi- 
tive ;  negative  ? 

36.  For  what  range  of  values  of  (3  between  0  and  2  rr  is  tan  jS  —  cot/3  posi- 
tive; negative? 

37.  Show  that  tan  y  +  cot  y  must  always  be  numerically  greater  than  unity. 

38.  Trace  the  variation  of  sin^  ^  as  ^  varies  from  0  to  2  tt.. 

39.  Trace  the  variation  of  cos^  ^  as  ^  varies  from  0  to  2  tt. 

40.  Trace  the  variation  of  1  —  sin  ^  as  ^  varies  from  0  to  2  tt. 

41.  Trace  the  variation  of  1  —  cos  ^  as  ^  varies  from  0  to  2  tt. 

42.  Find  by  inspection  logg  .625,  log8i27,  loggg  .008. 

43.  What  numbers  correspond  to  the  following  logarithms  to  base  9 :  —  3, 
-2,    -1.5,    -1,0,  .5,  1,2,3? 

44.  In  the  formula  W=^^^p^v^  [(  ~)"^  ~  "^T  ^^^^^  ^ives  the  work 
of  an  air  compressor,  find  W  when  n  =  1.3,  p^  =  14.7,  p2  =  72,   vi  =  6. 

45.  Work  the  following  with  the  slide  rule  : 

,.    .72x137x14      .     .J.    fl20y-^      .       .,   42  sin  27°      . 
^^^        372x778      ='     ^'M42J     =^      ^'^-13^-  =  ^ 

46.  Solve  for  a: :    52^  =  6;    8»=-i  =  7. 

Query.     Does  the  result  depend  on  the  base  of  the  system  of  logarithms 
used? 

47.  Solve  for  x:   32«  -  4  •  3''  +  3  =  0. 

48.  Find  the  amount  of  $2000  in  5  years  at  4%  compound  interest. 

49.  At  what  rate,  compound  interest,  will  $45,000  amount  in  8  years  to 
$60,000? 

50.  In  how  many  years  will  a  city  become  three  times  its  original  size  if 
it  increases  |  each  year  ? 


REVIEW   EXERCISES  123 

51.  Derive  the  formulas  of  Art.  42  from  those  of  Art.  40.     [Suggestion. 
Multiply  respectively  by  a,  b,  and  -  c  and  add.] 

52.  Derive  the  formulas  of  Art.  40  from  those  of  Art.  42.     [Suggestion. 
Solve  for  a  cos  /8  and  b  cos  a  and  add.] 

53.  From  the  law  of  sines,  Art.  41,  show  that  ^^^-^  =  sm^-smy^ 

b  +  c      sin  13  +  sin  y 

54.  By  applying  the  formulas  of  Art.  70  to  the  result  obtained  in  example 
53,  derive  the  law  of  tangents  of  Art.  48. 

55.  From  the  formulas  of  Arts.  42  and  68  derive  the  results 

sin  ^^  =  J5ZS5ZI) ;  COS  «  =  JiSZ3 ;  tan  ^  =  JKEMlZs)  . 

2       ^  be  '  2       >        6c  2       >      s(s-a) 

56.  Draw  the  graph  of  sin  ^  + cos  ^  and  thus  trace  its  variation.     What 
values  of  0  cause  the  given  expression  to  assume  maximum  values  ;  minimum? 

57.  Draw  the  graph  of   tan  0  +  cot  0  and  thus  trace  its  variation.     What 
values  of  0  make  the  given  expression  a  maximum ;  a  minimum  ? 

58.  Draw  the  graph  of  arcsin  u  and  trace  its  variation.      [Suggestion. 
Lay  off  the  values  of  u  as  abcissas,  of  arcsin  u  as  ordinates.] 

59.  Draw  the  graph  of  arccos  u  and  trace  its  variation. 

60.  Draw  the  graph  of  arctan  u  and  trace  its  variation. 

61.  Draw  the  graph  of  arccot  u  and  trace  its  variation.     What  discon- 
tinuities are  exhibited  by  the  functions  of  examples  58-61  ? 

62.  Find  from  the  table   the   values  of    cos  625°  12' ;   of  sin  238°  25';  of 
tan  324°  6';  of  cot  921°  32'. 

63.  Find  without  reference  to  the  table  the  value  of 

cos  285°  cos  345°  +  sin  195°  sin  465°. 

64.  Find  without  reference  to  the  table  the  value  of 

tan  205°  cot  335°  +  tan  295°  cot  115°. 

65.  Find   all  the   values   between    0   and  2  tt  of 

arcsin  (  —  )  ;    arccos  ( ] :  arctan  - . 

V     2/'  V      V2/  3 

66.  Find  all  the  values  of  a  between  0  and  2  tt  if 

sin  3  a  Z3  A  ;  if  tan  2  a=  -  \/3  ;  if  cos  -  =  —  ;  if  cot  -  =  -  1. 

V2  '2      V2  3 

67.  Find  the  value  of  sin  (2  a  +  /3)  if  a  =  arcsin  |  and  of  2d  quadrant, 
ft  =  arctan  |  and  of  1st  quadrant. 

68.  Find  the  value  of  tan  (3  a  +  2  /?)  if  a  =  arcsin  |  and    of  2d  quadrant, 
P  =  arccos  y\  and  of  4th  quadrant. 

69.  Derive  the  formula  for  sin  («  +  /8  +  y)  in  terms  of  sine  and  cosine. 

70.  Derive  the  formula  for  cos  (^  +  (S  -\- y)  in  terms  of  sine  and  cosine. 


124  REVIEW  EXERCISES 

71.  Derive  the  formula  for  tan  («  +  ^8  +  y)  in  terras  of  tangent. 
Discuss  the  results  of  examples  69-71  in  case  a  +  ft  +  y  =  ir. 

72.  In  the  results  of  examples  69-71  put  ft  =  y  —  a,  and  thus  obtain  the 
formulas  for  sin  3  a,  cos  3  a,  and  tan  3  a. 

73.  Find  the  value  of  cos  (3  a  —  2  /3)  if  «  =  arccos  (  —  jV)  and  of  3d  quad- 
rant and  ^  =  arctan  |  and  of  1st  quadrant. 

74.  Find  the  value  of  cot  {^  a  —  ft)  \i  a  —  arcsin  f  and  of  1st  quadrant 
and  ft  =  arccos  (-  ^f)  and  of  2d  quadrant. 

75.  Find  the  value  of  sec  (a  +  /8)  if  a  =  arctan  ^^  and  ft  =  arcsin  ^V?  both 
of  1st  quadrant. 

2  tan  a 


76.  Show  that  sin  2  = 

77.  Show  that  cos  2  a  -- 


1  +  tan'-^  a 
1  -  tan2  a 


1  +  tan2  a 

78.  Show  that  tan  ( 45°  —  ^  )  =:  esc  a  +  cot  a. 

79.  Show  that  cot  (45° —  - J  =csc  a  — cot  a. 

Transform  into  products  or  quotients  the    following  expres- 
sions (80-84) : 

80.  cot  a  +  tan  a. 

81.  cot  a  —  tan  a. 

82.  1  +  tan  a  tan  ft. 

83.  cot  a  —  tan  ft. 
g^     cot  a  +  cot  ft 

tan  a  +  tan  ft 

If  a  +  /3  +  7  =  TT,  show  that  (85-87) 

85.  sin  a  +  sin  /?  +  sin  y  =  4  cos  -  cos  ^  cos  ^.        [Suggestion.      Apply 

A         Ji        Ji 

Art.  70  to  the  first  two  terms  and  Art.  67  to  the  third  term.] 

86.  cos  a  +  cos  ft  +  cos  y  =  1  +  4  sin  ^  sin  ^  sin  ^  • 

Jd  Z  2i 

87.  tan  a  +  tan  ft  +  tan  y  =  tan  a  tan  y8  tan  y.     (See  example  71.) 

88.  How  does  it  appear  from  example  87  that  either  all  three  angles  of  a 
triangle  are  acute  or  else  two  are  acute  and  one  obtuse?     (Consider  the  signs.) 

89.  How  does  it  appear  from  example  87  that  if  one  angle  of  a  triangle 
is  obtuse,  it  is  numerically  nearer  90°  than  either  of  the  acute  angles? 

In  the  following  equations  find  the  angle : 

90.  tan  2  a  tan  «  =  1. 

91.  sin  (60°  -  ft)  -  sin  (60°  +  yg)  =  f . 


SECONDARY   TRIGONOMETRIC   FUNCTIONS  125 

92.  cos  6  y  —  cos  2  y  =  0. 

93.  r  sin  ^  =  8,  r  cos  ^  =  15. 

94.  r  sin  6  cos  <f>  =  S,  r  sin  6  sin  <^  =  4,  r  cos  6  =  12. 

95.  Show  that  sin  ^  sin  ^  sin  ^  sin  i^  =  A. 

5    5     5     5   16 

96.  Show  that  cos  —  cos  |^  cos  ^  cos  i^  =  -i;  • 

15  15  lo  15       lb 


97.  Show  that  2  sin  ^  =  ±  VI  +  sin  a  ±  VI  -  sin  a. 

98.  Show  that  2  cos  ^  =  ±  Vl  +  sin  a  =f  VI  -  sin  a. 

99.  The  formula  for  the  horizontal  range  of  a  projectile  fired  at  an  eleva- 

tion  a  with  a  muzzle  speed  u,  is  —  sin  2  a.     Show  that  the  maximum  range 

9 
is  attained  for  an  elevation  of  45*^. 

100.  A  triangle  is  formed  by  two  given  sides  of  constant  length  b  and  c, 
including  a  variable  angle  a.  For  what  value  of  a  is  the  third  side  a  maxi- 
mum ;  the  area  a  maximum? 


SECONDARY   TRIGONOMETRIC    FUNCTIONS 

In  addition  to  the  trigonometric  functions  defined  in  Art.  6, 
32,  and  54,  there  are  certain  other  expressions  which  are  also 
functions  of  the  angle.  While  of  less  importance  than  the  six 
primary  functions,  an  investigation  of  their  properties  will  be 
valuable,  not  only  for  the  results  obtained,  but  as  a  review  of  the 
fundamental  principles  of  trigonometry. 

We  may  define,  then, 

versed  sine  a  =  vers  a  =  1  —  cos  a, 
coversed  sine  a  =  covers  a  =  1  —  sin  a, 
exsecant  a  =  exsec  a  =  sec  a  —  1, 
excosecant  a  =  excsc  a  =  esc  a—  1, 


101.    By  reference  to 

Fig. 

53, 

show  that 

vers  a  =  1 , 

V 

covers  «  =  1  —  - 

V 

exsec  a  =  -  —  1 , 

excsc  a  =  ' —  1 . 

V 

102.  By  reference  to  Figs.  60-63,  show  that,  in  line  representations, 
vers  a  =  MA,  covers  a  =  NB, 

exsec  a  =  PT,  excsc  a  =  PS. 

103.  Determine  the   signs  and  limitations  in  value  of  each  of  the  four 
secondary  functions  in  the  different  quadrants. 


126  REVIEW   EXERCISES 

104.  Show  that  if  the  quadrant  of  the  angle  and  the  value  of  any  one  of 
its  ten  functions  are  given,  the  values  of  the  other  nine  can  be  found. 

105.  Find  the  values  of  the  four  secondary  functions,  given  : 

a  =  arcsin  (  —  ^%"j-)  and  of  '3d  quadrant ;  jS  =  arccos  |§  and  of  4th  quadrant ; 

y  =  arctan  (—  -^^j)  and  of  2d  quadrant;  8  =  arccot  ff  and  of  1st  quadrant. 

106.  Find  all  ten  functions,  given: 

a  =  arcvers  |°  and  of  4th  quadrant;  ^  =  arccovers  t|  and  of  2d  quadrant; 

y  =  arcexsec  ^  and  of  1st  quadrant ;  8  —  arcexcsc  2^^  and  of  3d  quadrant. 

107.  Trace  the  variation  of  each  of  the  secondary  functions  as  the  angle 
varies  from  0  to  2  tt. 

108.  Draw  the  graph  of  each  of  the  secondary  functions.     What  discon- 
tinuities, if  any,  are  present. 

109.  Find  the  secondarv  functions  oi  k  •-,  for  ^  =  1,  2,  •••  8. 

4 

110.  Find  the  secondary  functions  of  ^' .--,  for  ^  =  1,  2,  •••  12. 

111.  Verify  the  relations  of  Art.  61  for  the  secondary  functions. 

112.  Determine  the  relations  analogous  to  those  of  Art.  C2   affecting  the 
secondary  functions.     [Suggestion.     Use  Art.  60.] 

113.  By  means  of  the  cosine  series.  Art.  77,  Eq.  (.5),  show  that  lim  Y^^—  —  0. 

PXSPO  0 

114.  From  the  preceding  example,  show  that  lim  — '-^ —  =  0. 

115.  Show  that 

vers  a  +  covers  a      exsec  a  +  excsc  a       2  vers  a  covers  a 


vers  cc  —  covers  a      exsec  a  —  excsc  a      vers  a  —  covers  a 

116.  Show  that  exsec^  /?  +  2  exsec  fi  =  tan^  p. 

117.  Solve  and  find  y  in  degrees :  2  vers  y  (2  —  vers  y)  =  1. 

118.  Solve  and  find  8  in  degrees :  tan^  8  +  exsec  8  =  4. 

119.  Show  that,  if  a  is  the  angle  at  the  center  of  a  circle  of  radius  r,  the 
ordinate  at  the  middle  of  the  chord  is  given  by  the  formula  m  =  r  vers -.  Find 
m  for  r  =  1433,  a  =  11°  32'.  ^ 

120.  If  T  is  the  intersection  angle  of  two  tangents  to  a  circle  of  radius  r, 
the  shortest  distance  of  their  point  of  intersection  from  the  arc  is  given  by 

the  formula  d  =  r  exsec  -.     Find  d  for  r  =  5730,  t  =  5°  32'. 

121.  Reduce  the  first  member  to  the  second  in  the  identity 

(exsec  a  +  vers  a)  (excsc  a  +  covers  a)  =  sin  a  cos  a. 

122.  Show  that  vers  2  a  =  2  sin^  a. 

123.  Show  that  exsec  2  «=  _2_sm^-a__^ 

1-2  sin2  a 

124.  Show  that  excsc  2  a  cos  2  a  =  tan  a. 


REVIEW   EXERCISES 


127 


125.  At  points  in  a  straight  line  ordinates  are  erected  siich  that  for  each 
point  (x,  y),  x  =  vers  (arcsin  y).  Show  that  the  graph  thus  determined  is  a 
circle  tangent  to  the  F-axis  at  the  origin. 

126.  At  each  point  in  a  circular  arc  the  radius  is  extended  an  amount  equal 
to  the  exsecant  (in  line  values)  of  the  arc  measured  from  a  fixed  point  in  it. 
What  is  the  graph  thus  determined? 

127.  Two  equal  circles  have  their  centers  in  the  same  horizontal  line.  Show 
that  the  horizontal  distance  between  two  points  in  the  neighboring  arcs  is  equal 
to  twice  the  versed  sine  (in  line  values)  of  the  arc,  measured  from  the  point  of 
tangency. 

128.  Two  tangents  to  a  circle  intersect  at  an  angle  t.  Show  that  the  dis- 
tance of  the  point  of  intersection  from  the  midpoint  of  the  chord  of  contact 
equals  exsec  t  +  vers  t  (in  line  values). 

129.  Show  how  the  versed  sine  is  of  practical  use  in  staking  out  a  circular 
railroad  track  passing  through  three  given  points. 

130.  Show  how  the  exsecant  is  of  practical  use  in  staking  out  a  circular 
railroad  spur  of  given  radius  branching  tangentially  from  a  straight  track. 

Compute  the  missing  parts  of  the  following  triangles,  distinguishing  right 
from  oblique : 


"A 

'3 

a 

b 

c 

131. 

37°  42.8' 

90° 

6244.8 

132. 

72°  25.6' 

90° 

64.863 

133. 

90° 

375.84 

296.57 

-  134. 

54°  36.9' 

24.465 

42.850 

^135. 

136°  36.8' 

36902 

37490 

136. 

68°  51.5' 

90° 

7532.8 

137. 

90° 

396.45 

531.53 

138. 

90° 

.005428 

.006395 

139. 

148°  24' 

7.4536 

5.3648 

140. 

.038456 

.028638 

.051524 

141. 

125°  34.6' 

35°  25.3' 

2584.6 

142. 

24°  36.8' 

2.4657 

3.6542 

143. 

80°  04.5' 

90° 

30.007 

144. 

94°  46.8' 

34.086 

52.475 

145. 

.93274 

.40586 

.63208 

146. 

76°  46.3' 

85°  38.7' 

8.4637 

147. 

90° 

29846 

53857 

148. 

29°  57.4' 

43°  52.6' 

64.475 

149. 

17°  46.8' 

.39475 

.29478 

—  150. 

36875 

28467 

48542 

128  REVIEW   EXERCISES 

151.  In  a  given  triangle  a  =  280,  c  =  420,  y  =  38° ;  find  the  radius  of  the 
circumscribed  circle. 

152.  In  a  given  circle  a  =  63,  &  =  81,  y  =  54°;  find  the  lengths  of  the  bisec- 
tors of  the  interior  angles. 

153.  The  sides  of  a  given  triangle  are  220,  350, 440 ;  find  the  lengths  of  the 
three  medians. 

154.  In  a  given  triangle  b  =  340,  a  =  48°,  y  =  63°;  find  the  lengths  of  the 
radii  of  the  inscribed  and  of  the  three  escribed  circles. 

155.  A  boat  drifts  in  a  stream  whose  current  runs  4  miles  an  hour  due  east 
under  a  breeze  of  10  miles  an  hour  from  the  southwest.  Determine  the  motion 
during  35  minutes,  if  the  resistance  reduces  the  effect  of  the  wind  30  %. 

156.  Three  forces  of  1800,  2200,  and  2700  dynes  are  in  equilibrium ;  find 
the  angles  they  make  with  one  another. 

157.  A  helical  spring  is  fastened  to  the  door  16  inches  from  the  axis  of  the 
hinges,  and  to  the  jamb  4  inches  from  the  same  line  in  the  same  horizontal 
plane.  Find  the  length  of  the  spring  when  the  door  is  closed,  open  at  30°,  45°, 
70°,  90°,  120°.     Neglect  the  thickness  of  the  door. 

158.  A  cable  30  feet  long  is  suspended  from  the  tops  of  two  vertical  poles 
20  feet  apart  and  15  and  18  feet  high,  and  bears  a  load  of  200  pounds  hanging 
from  it  by  a  trolley.  Find  the  position  of  the  trolley  when  at  rest,  and  the 
lengths,  inclinations  to  the  horizon,  and  (common)  tensions  of  the  segments  of 
the  cable.     Neglect  the  weight  of  the  cable. 

159.  Let  the  data  be  as  in  the  preceding  example,  save  that  the  load  hangs 
from  a  ring  knotted  at  the  center  of  the  cable ;  find  the  inclinations  to  the  hori- 
zon and  the  (unequal)  tensions  of  the  segments.  Solve  when  the  ring  is 
knotted  at  a  point  12  feet  from  the  lower  end  of  the  cable. 

160.  The  eye  is  40  inches  in  front  of  a  mirror  and  an  object  appears 
to  be  35  inches  back  of  it,  while  the  line  of  sight  makes  an  angle  of  48°  with 
the  mirror.  Find  the  distance  and  direction  of  the  object  from  the  eye. 
(Note.     The  angles  of  incidence  and  reflection  are  equal.) 

161.  The  line  from  the  eye  to  the  object  recedes  from  the  mirror  at  an 
angle  of  32°,  while  the  object  is  36  inches  from  the  eye  and  12  inches  from  the 
mirror.     Find  the  angles  of  incidence  and  reflection,  and  the  point  of  reflection. 

162.  Two  railway  tracks  intersect  at  an  angle  of  75°,  and  are  connected  by 
a  circular  "  Y  "  of  800  feet  radius  lying  in  the  obtuse  angle  and  tangent  to  the 
two  tracks.  Find  the  distances  of  the  points  of  tangency  from  the  crossing  and 
the  length  of  the  "  Y  ". 

163.  Two  railway  tracks,  intersecting  at  an  angle  of  62°,  are  joined  by  a 
circular  "  Y  "  in  the  acute  angle  and  tangent  to  the  two  tracks  at  points  900 
feet  from  the  crossing.     Find  the  radius  of  the  "  Y  "  and  its  length. 


REVIEW   EXERCISES 


129 


Fig.  72. 


164.  In  setting  a  door  frame  6  feet  wide  and  8  feet  high,  the  vertical  side  is 
found  to  be  2  inches  (horizontally)  out  of  plumb.  Find  the  angles  of  the  paral- 
lelogram and  the  lengths  of  the  diagonals.  Is  the  diagonal  of  the  true  rectan- 
gle the  average  (arithmetic  mean)  of  these  two  ? 

165.  The  staking  out  of  a  certain  building  requires  the  setting  of  stakes  at 
the  vertices  of  a  rectangle  32  x  48  feet.  A  test  of  the  trial  setting  shows  the 
figure  to  be  a  parallelogram  whose  sides  are  as  given  above,  but  whose  diagonals 
dilfer  by  9  inches.  Find  the  angle  through  which  the  longer  sides  must  be 
swung  to  correct  distortion,  and  the 
chord  of  the  arc  through  which  the 
back  corners  must  be  moved. 

166.  A  triangular  roof  truss  is  80 
feet  long  and  divided  into  8  equal  seg- 
ments, by  vertical  members,  as  shown 
in  Fig.  72.     The  height  being  25  feet,  what  are  the  lengths  and  inclinations 
to  the  horizon  of  the  various  members? 

167.  The  triangular  roof  truss  shown  in  Fig.  73  is  60  feet  long  and  20 
feet  high.  The  bottom  chord  and  rafters  are  divided  into  equal  segments. 
Find  the  lengths  and  inclinations  of  the  members. 

168.  In  order  to  determine  the 
exact  location  of  the  point  G  (Fig. 
74),  a  base  line  AB  \s  laid  oif  due 
north  and  south,  measuring  precisely 
130  rods.  Convenient  intermediate 
stations  are  chosen,  and  angles  meas- 
ured as  follows:  ^^0=^36°  35', 
BA C  =  61°  10',  BAD  =  43°  54',  CAD  =  17°  16',  DCE  =  66°  36',  CDE  =  47°  41', 
EDF=55°4:H',DEF  = 
73°  12',  FEG  =  5S°32\  B 
EFG  =  10°  28'.  Com- 
pute the  distances  com- 
posing the  sides  of  the 
triangles  in  the  figure. 

169.  By  projecting 
the  distances  AC^  CE, 
EG  (Fig.  74),  perpen- 
dicular and  parallel  to 
A  B,  compute  the  east- 
erly and  southerly  dis- 
tances   of    G    from    A; 

find   thence   the    direct  Fig.  74. 

distance  and  direction  of    G  from  A. 


Fig.  73. 


FORMULAS 

GENERAL  TRIGONOMETRY 

1 


CSC  a  — 

sec  «  = 

cot  a  — 

tan  a  = 

cot  a  = 

sill  a 

sin^a  +  cos^  a  =  1. 

tan^  a  +  I  =  sec^  a. 
cot^H-  1  =  csc2  a. 
2  TT^  =  360°. 
F(2k7r  +  a)  =  F(a),  k  an  integer. 


sm  a 

1 

cos  a 

1 

tan  a 

sin  « 

cos  a 

cos  a 


Ffk 


±  a]  =  ±F (a),  k  an  even  integer. 


Fik'-±  a]=  ±  co-F (a),  ^  an  odd  integer, 

sin  (a  ±  /3)  =  sin  a  cos  yQ  ±  cos  a  sin  yS. 
.    cos  (a  ±  y6)  =  cos  a  cos  /3  =F  sin  a  sin  /3. 

tan(«±^)=   tan«±tanff  ^ 
1  =F  tan  at'dn  8 


cot(«±ff)=«"t«C"t/3Tl. 
cot  y8  ±  cot  a 


130 


GENERAL   TRIGONOMETRY  131 

sin  2  a  =  2  sin  a  cos  a. 
cos  2  a  =  cos^  a  —  sin^  a  =  2  cos^  a  —  1  =  1  —  2  sin^  «, 

2  tan  a 


tan  2  a  = 
cot  2  «  = 


1  —  tan^  a 
cot^  ce  —  1 


2  cot  a 
sin  \a=  V|  (1  —  cos  a). 


cos  I  a  =  V|  {\  4-  cos  a). 

.1  /l  —  cos  a      1  —  cos  a 

tan  -  a  =  \— =  — : 

2  ^  1  H-  cos  a  sm  a 

,1  /I  +  cos  «      1  +  cos  a 

cot  -  a=V— ^- =— ^^ 

2  ^  1  —  cos  a  sm  a 

sin  a  cos  yS  =  |  [sin  (ct  +  y8)  +  sin  (a  —  /8)]. 

cos  «  sin  /3  =  J  [sin  (a  -f-  ^)  —  sin  (a  ^  /S)] . 

cos  a  cos  yS  =  I  [cos  («  +  /S)  +  cos  (a  —  6)] . 

sin  a  sin  /3  =  —  -|  [cos  (ot  +  yS)  —  cos  (a  —  ^)]. 

sin  a  cos  «  =  |^  sin  2  a. 

cos^  a  =  1  (1  +  cos  2  a). 

sin^  a  =  1  (1  —  cos  2  cj). 

sin  a  -\-  sin  p  =  2  sin  — — ^  cos  — --^  • 
sin  a  —  sm  /3  =  2  cos  — ^— ^  sm  — --^» 
cos  a  +  COS  p  =  2  cos  — —-^  cos  — — ^  • 


/o  o-     a-\-  S    .     a—  ^ 

cos  a  —  cos  p  =  —  2  sin  — ^  sm  — —^ 


RIGHT  TRIANGLES 

6^2  _|.  ^2  ^  ^, 

a  +  ^  =  90°. 

a       .  ^ 

-  =  sin  a  =  cos  p. 


132 


FORMULAS 


=  cos  a  =  sin  y8. 


a     . 

-  =  tan  a 

0 


COtyS. 


A=  ^  ab  =  ^bcsin  a=  I  c^  sin  a  cos  a  =  ^  c^  gj^  2  a. 


OBLIQUE   TRIANGLES 

«  +  ye  +  7=180°. 

(?  =  5  cos  a-\-  a  cos  yS,  etc. 

a     _     h     _     <? 
sin  ct      sin  yS      sin  7 

c^  =  ^2  _j_  j2  _  2  a5  cos  7,  etc. 

tan  ^  ^  ^  = cot  -,  etc. 

2  0  -\-  c        2 


tan 


,  etc. 


2      s-a 


r  =  ■%  /('^  — ^)(g— ^)(g—  g) 


.A=j^  5(?  sin  a  =  rs  =  Vs  (s  —  a)  (s  —  6)  (s  —  (?). 


RIGHT   SPHERICAL  TRIANGLES 

sin  a  =  sin  <?  sin  a. 
sin  6  =  sin  c  sin  /3. 
tan  6  =  tan  c  cos  a. 
tan  a  =  tan  <?  cos  0, 
tan  a  =  sin  h  tan  a. 
tan  b  —  sin  a  tan  yS. 
cos  c  =  cos  a  cos  5.' 
cos  /3  =  cos  b  sin  a. 
cos  a  =  cos  a  sin  /?. 
cos  (?  =  cot  a  cot  /3. 


GENERAL   TRIGONOMETRY 


133 


OBLIQUE   SPHERICAL  TRIANGLES 

sin  a       sin  h       sin  c 


sin  a      sin  /3      sin  7 

cos  (?  =  cos  a  cos  J  +  sin  a  sin  5  cos  7,  etc. 

cos  7  =  —  cos  a  cos  y8  +  sin  a  sin  y8  cos  c,  etc. 


tan  -  = 


tan  r 


2      sin  (s  —  a) 


,  etc. 


tan  r 


_     /sin  (g  —  a)  sin  {s  —  h)  sin  (s  —  g) 
^  sin  s 


tan  -  = 


a  _  cos  (^S  —  a ) 


,  etc. 


2  cot  Z^ 

^=l(«  +  ^  +  7). 


cot  i2  =  J  -  c<>^  (>S^-  «)  cos  (S-fi)  cos  (a^-  7) 
^  cos  aS' 

tan  1  (/3  +  7)  =  ^^^i^l^cot  1  a,  etc. 
cos  -^-  (0  +  c) 


tan  1  (5  -  ^)  =  ^"'  2  (f — ll  tan  1 «,  etc. 
2^  sin  1(^4- 7) 

tan  1  (5  +  c?)  =  ^"^t  ^^~^^  tan  J  a,  etc. 
-  V  cos  1  (^  -  7) 


ANALYTIC  TRIGONOMETRY 


lim 

0=0 

lim 
0=0 


'  e  1 

_sin  ^J 

•  e  - 

tan  ^ 


=  1. 


=  1. 


(cos  a-\-{  sin  a)  (cos  ^+  i  sin  /3)  =  cos  (a  +  /3)  +  i  sin  (a  +  /3). 
(cos  «  +  i  sin  a)*^  =  cos  na  4-  ^  sin  na. 
cos  a  +  ^  sin  a  =  e**". 
cos  «  —  I  sin  a  =  e~**. 

COS  a  = • . 


134  ANALYTIC   TRIGONOMETRY.     CONSTANTS 

sm  a  =  — — 

2^ 

cosh  a  +  sinh  a  =  g*. 

cosh  a  —  sinh  a  =  e~"-. 


cosh  «  = 
sinh  a  = 


2> 


iog(i+.)=f-|Vf-|V-. 

C0S«=l--  +  jj-^+-. 


tan  «  =  -  +  —  +  — ^  + 


1      8      15       315 

cosh.=  l  +  -  +  -+^4 


CONSTANTS 

7r  =  3.14159265  •••. 

7r^  =  180°. 

l^  =  5T.295779e5°  •••  =57°  17'44.8'^... 

g  =  2.7182818285  ••.. 

Mode  10  =  — ^  =  .4342944819  .... 


CONSTANTS  135 

1  inch  =  2.54001  •••  centimeters. 
1  foot  =  .3048  .-.  meters. 
1  mile=  1.60935  •••  kilometers. 
1  centimeter  =  .3937  •••  inches. 
1  meter  =  3.28083  feet  =  1.09361  yards. 
1  kilometer  =  .62137  miles. 

g=  32.086528  +  .171293  sin^t^  feet  per  second  per  second. 
=  9.779886  +  .05221  sin^  (^  meters  per  second  per  second  at 
sea  level  for  latitude  ^. 


ANSWERS   TO   EXERCISES 

(Answers  are  omitted  in  case  their  knowledge  would  detract  from  the  value  of  the 

exercise.) 

Exercise  II 

6.  (0,   0),    (a,   0),    (a,   a),  (0,   a);     (^  V2,  o),    ("' |  Vs),    (-|V2,o), 
(0,-|V2). 

7.  (5,  0),  (0,  -  5),  (-4.33,  -2.5),  (-3.54,  3.54). 
9.   5.6569;  7.6158. 

11.  Cross  country  distances,  in  miles :  5.099;  2.828;  2.236;  2.236;  6.325. 

12.  Distances  saved,  in  yards  :  773.2  ;  644.4 ;  1288.7 ;  128.9. 


Exercise  IV 

9. 

1681 

11. 

If. 

13.    cos 

a. 

15.  Jl-siny 
^l  +  sin  y 

1519 

12. 

If. 

14.    2csc^. 

10. 

f. 

Exercise  V 

16.  2(l+tan2y). 

9. 

h 

13. 

60°. 

17. 

(a) 

60°;    (b)  19.05  ft. 

10. 

W3. 

14. 

0°  and  60°. 

18. 

8.08 

;  16.17. 

11. 

i- 

15. 

60°  and  90°. 

19. 

452.39. 

12. 

K3V3- 

-2). 

16. 

45°. 

20. 

60°. 

Exercise  VI 
11.   0°  and  .60°.        12.    30°.        13.   0°,  30°,  and  45°.         14.   0°,  30°,  and  45°. 

Exercise  VII 

1.  (a)  6.7,6.7;  (b)  8.23,4.75.  4.    15.6;  9. 

2.  (a)  8;  (6)   13.86.  5.    (a)  2598.16;  (b)  1500. 

3.  (a)  20,  34.64 ;  (6)  28.28,  28.28.      6.   24  miles  per  hour,  30°  east  of  north. 

'     Article  18 

1.  a  =  40.32,  &  =  11.76.  3.    6  =  151.5,  c  =  381.6. 

2.  a  =  20.25,  c  =  33.75.  4.    a  =  133.2,  b  =  149.2. 

137 


138  ANSWERS   TO   EXERCISES 

Exercise  VIII 

(These  results  were  ol)tained  with  four-place  tables.) 

1.  ^  =  64°  50',  a  =  14.46,  h  =  30.77. 

2.  p  =  37°  40',  a  =  57.79,  b  =  44.61. 

3.  a  =  28°  45',  a  =  116.88,  b  =  213.04. 

4.  a  =  11°  25',  a  =  103.11,  6  =  510.68. 

5.  (3  =  68°  35',  b  =  599.13,  c  =  643.66.     • 

6.  /3  =  17°  15',  b  =  223.56,  c  =  753.93. 

7.  a  =  9°  30',  b  =  7170.96,  c  =  7272.73. 

8.  a  =  72°  30',  a  =  4757.40,  c  =  4988.36. 

9.  a  =  41°  49',  /3  =  48°  11',  6  =  268.33. 

10.  a  =  32°  12',  (3=  57°  48',  6  =  605.03. 

11.  a  =  34°  13',  13  =  55°  47',  a  =  354.25. 

12.  a  =  53°  8',  ^  =  36°  52',  a  =  1120. 

13.  a  =  36°  52',  ^  =  53°  8',  c  =  1080. 

14.  «  =  44°46',   )8  =  45°14',    c  =  845.07.  ' 

15.  a  =  29°  11',   ^  =  60°  49',   c  =  440.94. 

16.  a  =  59°  41',  13  =  30°  19',  c  =  2445.55. 

17.  a  =  29°  29',  (3  =  60°  31',  b  =  168.00,  c  =  193.00. 

18.  a  =  41°  04',  f3  =  48°  56',  a  =  230.00,  c  =  350.13. 

19.  (3  =  15°  40',  a  =  93.47,  b  =  26.21,  c  =  97.08-. 

20.  a  =  65°  10',  a  =  60.35,  6  =  27.93,  c  =  66.50. 

21.  200.1  ft.  23.   2°  23'.  25.   1°  9'. 

22.  1501.73  ft.  24.   4°  46'.  26.   33°  41',  26°  34',  45°. 

27.    .134  pitch,  .2887  pitch,  |  pitch. 

28.  19°  28'  inclination.       31.    26°  31'.  34.  21.73  ft.  =  21  ft.  8|  in. 

29.  8°  3'  inclination.  32.    5859.71  ft.  35.  260.4  ft. 

30.  0°  9',  0°  17',  1°  26'.       33.   32°  28'.  36.  0°  20'. 

37.  5°  54'  (  =  .1029  radians).  Note  that  .1029  =  sin  5°  52.5',  an  approxima- 
tion.    See  Arts.  72  and  77  (3). 

38.  2.468  miles.  39.   502.2  ft. 

40.  ^=(0,0),  5  =  (240.9,0),  C=  (385.9,  274.8), 
D  =  (98.7,  814.6),       E  =  (162.8,  1043.0),         F=  (649.1,  1248.8). 

41.  13  =  110°,  b  =  68.81,  A  =  828.81;  ^  =  36°,  a  =  202.27,  A  =  12,641.88; 
a  =  75°  06',  p  =  29°  48',  A  =  30,441.6  ;  a  =  63°,  b  =  326.88,  A  =  52,425.01  ; 
a  =  64°  17',  a  =  553.12,  A  =  119,594.88;  a  =  41°  1',  ^  =  97°  58',  A  =  202,809.6. 

42.  51.76,   61.80,   68.40,  76.54,   1,  121.76,   141.42,   173.20. 

43. 
45. 


5  =  2scos?. 
4 

44. 

s  =  2Rsin''''  =  2rt.u'''' 
n                       n 

n 

2'7rR 

cos  180°. 
n 

e  =  62.83,  P4=  56.57,     0^=  44.43; 

P8  =  61.23,     C8  =  58.05; 

P,e  =  62.43,   c,«=  61.62; 

P32  =  62.72,   C32  =  62.53. 

Ap 

^nW^&ui 

180°        180°      1     po  .    360° 
cos =  -  nR^  sm 

n              n         2 

n 

Ai 

=  tt/^^  cos 

n        2          \ 

360°\ 

,     ^P4 

=  200.00, 

.4,4  =  157.08; 

Ap^ 

=  282.84, 

^,-8  =  268.15; 

^P16 

=  306.16, 

A^Q  =  302.21 ; 

^P32 

=  312.16, 

^,32  =  311.14. 

^6- 

60.00,     c,. 

=  54.41 ; 

^2  = 

62.11,    c,. 

=  60.69 ; 

^^24  = 

62.64,    c.„ 

=  62.29 ; 

^48  = 

62.78,   c,« 

=  62.69. 

1,     ^^6  =  259.80, 

J,g=  235.62; 

Aj,^^  =  300.00, 

.4  ,-,2  =  293.11; 

^,24  =  310.56, 

^/  =  308.80; 

A,^  =  313.20, 

^,-^8  =  312.81. 

ANSWERS   TO   EXERCISES  139 

46.  Ac=7rR% 

^,.  =  314.16, 

47.  C  =  62.83, 

48.  ^.  =  314.16 


49.  Z  =  -30,    F=- 17.321,   22  =  34.641,  30° south  of  west. 

50.  A'  =  6  r,    F  =  0,    R  =  6  r,    due  east. 

51.  Distance  from  center  =  r  cos  0.  52.    x=  —  15. 

53.  Component   along   plane  =  g  sin  a,  component  perpendicular  to   plane 
=  g  cos  a. 

54.  16,   27.71;   8.28,  30.91;   5.56,   31.51;   2.79,   31.88. 

55.  50  pounds  pressure,    141.42  pounds  along  ladder. 

56.  Z  =  57.28,    y=  30.73.         57.   i2  =  18.44,  ^  =  49°  24'.         58.   c=  11.99. 

Exercise  IX 

1.  4,1.5,  -2.5.  4.    1,8,  16,  i,^V 

2.  4,  if.  5.    (a)  .4724;  (b)   .01614. 

3.  1,  4,  16,  32,  64,  tV,  ^?-  6.    (a)  28.16;   (b)  .01913;   (c)  2.465. 

7.  17.978.  9.   524.9.  11.   a:  =  1.79.     13.    $4136.09.     15.    11.6  years. 

8.  .76252.         10.   a;  =  2.29.     12.    $4656.20.    14.    5.2%.  16.    3.8  years. 

Exercise  X 

1.  4.86024,  2.79187,  9.84198,  5.80872  -  10,  21.47712. 

2.  4.96088,  1.15518,  11.50651,  5.89510  -  10,  24.30103. 

3.  516.35,  4.0966  x  10^2,  .016335. 

4.  16361,  5.64325  x  10",  .00013671. 

5.  9,067,800,000.  7.   88.594  cm.  9.    13,231  x  lO^o. 

6.  7.0048  X  1010  cm.  8.  71.68  cm.  10.   2,754,100. 

11.  9.63459  -  10,  9.52928  -  10,  0.01824.  13.   78°  01.1',  81°  43.7',  76°  17.1'. 

12.  9.97454  -  10,  9.78340  -  10,  0.04197.  14.   25°  20.7',  27°  32.6',  35°  3.6'. 
15.   13.861.                  16.   .91186.                  17.   3.9968.  18.    .38875. 

19.  1.3365  inches.  22.    .074765. 

20.  .1111  foot  (=1.3332  inches).  23.    6:711;  8.381. 

21.  1.7%  less  than  the  true  value.  24.    -  11.85;   -  61.38. 


140 


ANSWERS   TO   EXERCISES 


Exercise  XII 

1.  ^  =  27^  a  =  2302.3,  h  =  1173.1. 

2.  cc  =  60"  37.6',  /3  =  29'^  22.4',  h  =  4238.9. 

3.  ^  =  14°  44.8',  b  =  254.07,  c  =  998.12. 

4.  a  =  15°  39.6',  ^  =  74°  20.4',  b  =  168.36,  c  =  174.85. 

5.  a  =  50°  13.1',  (3  =  39°  46.9',  c  =  9.5378. 

6.  ^  z:.  71°  34.5',  a  =  10.417,  ^>  =  31.271. 

7.  a  =  83°  38.4',  y8  =  6°  21.6',  6  =  14.82,  c  =  133.79. 

8.  a  =  75°  33',  ^  =  14°  27',  c  =  54.953. 

9.  a  =  64°  48.5',  /S  =  25°  11.5',  6  =  31,037. 

10.  ;8  rr  60°  9.8',  a  =  5.854,  c  =  11.766. 

11.  /?  =  64°  42.6',  a  =  19.023,  b  =  40.264,  c  =  44.531. 

12.  a  =  28°  23.6',  (3  =  61°  36.4',  c  =  .00042. 

13.  a  =  26°  47.3',  a  =  3.2196,  &  =  6.3769. 

14.  a  =  38°  23.3',  ft  =  51°  36.7',  a  =  .056677. 

15.  a  =  54°  43.6',  b  =  .44535,  c  =  .77120. 

16.  a  =  54°  43.2',  a  =:  242.79,  b  =  343.16,  c  =  420.37. 

17.  a  =  55°  59.3',  13  =  34°  0.7',  c  =  .0074192. 

18.  a  :=  9°  47.5',  a  =  .89928,  c  =  5.2878. 

19.  a  =  63°  20.7',  /3  =  26°  39.3',  a  =  .014523. 

20.  a  =  64°  41.8',  a  =  1563.4,  6  =  739.12. 


21. 
22. 


8.2583  feet. 

18  feet  3.8  inches. 


23.  141.42  square  feet. 

24.  60°  1.8'. 

25.  1237.8  feet. 

26.  16|  miles,  36°  52.2'  north  of  west. 

29.  nR^ sin'-^cos'^- 

n  n 

30.  2177.4. 


31. 
32. 
33. 
34. 
35. 
36. 

37. 

38. 


178.8  miles. 
0°  32'. 

.078523  feet. 
14834  feet. 
425.64  feet. 
142.4  feet. 

118.1  feet. 

554.06;  145.17. 


Article  47 

1.  a  =  33°  19.9',  a  =  438.23,  c  =  788.58. 

2.  a  =  65°  49.8',  a  =  122.13,  b  =  885.60. 

3.  (3  =  15°  57.0',  b  =  5.442,  c  =  17.865. 
—  4.  ^  =  1°  02.0',  a  =  9.368,  b  =  .18134. 


Article  48 

3.  a  =  57°  .59.9',  y  =  23°  36.6',  c  =  29.526. 

4.  ^  =  13°  55.6',  y  =  35°  30.7',  b  =  135.96. 
^      (a  =  104°  31.3',  13  =  40°  2.9',  a  =  5889.9 ; 

^    -      4°  37.1',  ^'  =  139°  57.1',  a'  =  489.8. 
94°  17.9',  y  =  47°  13.3',  a  =  207,810; 


6.    ^ 


(a 


[  a'  =  47°  4.6',  y'  =  132°  46.7',  a'  =  152,600. 


ANSWERS   TO   EXERCISES  141 


Article  49 


1.  ^  =  23°  42.8',  y  =3o°  45.2',  a  =  450.35. 

2.  a  =  23°  31.8',  y  =  19°  7.2',  h  =  818.54. 

3.  a  =  33°  17.5',  y  =  63°  13.1',  b  =  .11496. 

4.  ^  =  66°  27.0',  y  =  45°  11.2',  a  =  .005202. 


Article  50 


66= 

'49.4', 

y  = 

z6r 

*13.4'; 

A  = 

1.9181  X 

101 

42° 

'51.8', 

y  = 

=  54= 

•51.8'; 

A  = 

4.4175  X 

10^. 

34° 

45.4', 

7  = 

=  88^ 

'45.8'. 

72= 

'  33.2', 

^y-- 

=  QV 

^33.4'. 

r? 

7. 

Impossible. 

Why? 

8. 

y  =  8< 

'  58.3' 

'. 

1.  a  =  51°  57.2',  y8 

2.  a  =  82°  16.4', /3 

3.  a  =  56°  28.8',  ^ 

4.  «  =  45°  53.4',  13 

5.  Impossible.     Why? 

6.  IS=  136°  39.8'. 


Exercise  XIV 

1.  /8  =  74°  0.3',  y  =  43°  24.7',  a  =  76,568. 

2.  a  =  52°  56.6',  y8  =  79^^  47.8',  y  =  47°  15.6'. 

3.  y  =  86°  10.3',  b  =  8.4172,  c  =  9.0436. 

4.  a  =  43°  29.3',  y  =  80°  35.3',  6  =  .30470. 

5.  Impossible.     Why? 

6.  a  =  52°  11.2',  y  =  27°  38.8',  a  =  49,921. 

7.  a  =  34°  32.1',  ^  =  51°  41.8',  y  =  93°  46.1'. 

8.  y  =  69°  28.5',  a  =  67,439,  c  =  72.037. 

9.  a  =  23°  34.1',  jS  =  35°  35.7',  c  =  6.0804. 

10.  a  =  15°  35.2',  y  =  126°  7.6',  c  =  66.113. 

11.  (3  =  13°  11.7',  y  =  16°  24.1',  a  =  .082764. 
I  ^  =  32°  8.5',  y  =  89°  45',  b  =  34.993  ; 

■  i  /?'  =  31°  38.5',  y'  =  90°  15',  b'  =  36.210. 

13.  a  =  34°  11. .5',  a  =  382.48,  c  =  641.52. 

14.  a  =  28°  57.0',  /?  =:  104°  28.6',  y  =  46°  34.4'. 

15.  a  =  21°  13.9',  y  =  32°  19.7',  b  =  .0048578. 

16.  a  =  162°  18.9',  y  =  7°  08.3',  a  =  61.896. 

17.  a  =  33°  33.1',  /?  =  50°  42.0',  y  =  95°  44.9'. 

18.  a  =  75°  0.2',  a  =  8355.2,  b  =  6470.6. 

19.  a  =  45°  29.5',  /S  =  14°  15.5',  6  =  2146.7. 

20.  a  =  49°  36.8',  ^  =  40°  23.2',  c  =  952.67. 

21.  a  =  151°  56.6',  /8  =  4°  30.4',  y  =  23°  38.0',  b  =  416.45. 

22.  a  =  80°  0.0',  ^  =  54°  45.2',  y  =  45°  14.8',  a  =  124.81. 

23.  /?  =:  90°  50',  y  =  16°  0',  a  =  720.81,  c  =  207.58. 

24.  a  =  95°  26.6',  y  =  27°  8.4',  a  =  125.81,  b  =  106.49. 

25.  2.1815  X  109;  5.0105;  1,742,040,000. 

26.  2.567  X  109;  .038051;  7270.3. 

27.  Case  III. 


142 


ANSWERS   TO   EXERCISES 


28. 

29. 
30. 


a  =  15°  45',     yS 

a  =  2r  52.6  ,  fi 

a  =  91°  54.7',  13 

a  =  47°  59.5',  (3 
r  A  =  156°  55.6',  B  =  145°  57.2',  C  =  57°  7.2' ; 
t  ^  =  145°  13.6',  B  =  121°  11.4',  C  =  93°  35.0'. 


29°  15', 

c  =  52.1 ; 

42°  7.4', 

c  =  723.6; 

53°  5.3', 

c  =  43.042 ; 

72°  0.5', 

c  =  291.38. 

(A 


31.    ^ 


63°  25.9',  B  =  141°  34.1',  c  =  328.4 ; 
122°  25.4',  C  =  137°  34.6',  c  =  575.41. 


75°,  &  =  878.48,  cz=  621.17; 
L  A  =  113°  58.6',  B  =  106°  2.2',  C  =  139°  59.2'. 

32.  a  =  h  =  48.5. 

33.  ^  =  151°  2.7',  B  =  133°  25.9',  C  =  75°  31.4'. 

34.  d^  =  0.2  +  ^2  +  c2  _  2  a&  cos  ah -2  be  cos  kr  +  2  ac  cos  (a6  +  k-)  ;  d  =  12.98 

35.  be  =  84°  03.5',         cS  =  75°  53.0',         (/a  =  82°  03.5',         ab  -  cd  =  109.28, 
6c  -  da  =  114.47. 

36.  c  =  1001.1,  (/  =  568.6.  37.    «  rz  36°,  s  =  15.217. 
„„   6  - 


38.    V  = 


6 


30  (6  -  h)  \/l2  A  -  A- :   F^  =  135.0,  V^  =  371. 


F3  =  663.3,  F4  =  990.0,  F5  =  1338.0,  F^  =  1696.4,  F^  =  2054.8,  Fg  =  2402.8, 
F9  =  2729.5,  Fio  =  3021.1,  F„  =  3257.8,  V^^  =  3392.8. 
39.   6  =  483.4.  40.   a  =  3221.5.  41.   b  =  1286. 

42.  Distance  =  31.63,  total  height  =  20.97. 

43.  Distance  =  24.24,  height  =  5.08. 

44.  AD  =  738.2,  DB  =  150.6. 

45.  AC  =  1075.1,  BC  =  679.5. 

46.  ^D  =  1460,  DC  =  678,  angle  BXC  at  left  =  17°  27.5'. 

47.  Distance  =  135.74,  height  =  36.602. 

48.  D  =  57°  40',  CD  =  196.73,  BD  =  233.55. 

49.  AD  =  603.94,   ^C  =  693.12,  BC  =  838.82,  5^  =  595.76,  AB  =  867.48, 
angle  CXB  at  left  =  4°  27.7'. 

50.  AC  =  730.17,  ^Z>  =  737.37,  BE  =  805.40,  BF  =  715.52,  ^5  =  841.67. 

51.  39°  54'. 

Exercise  XV 

1.   45°,  60°,  150°,  112°  30',  171°  53'  14.4",  42°  58'  18.6". 


2. 


77-^  TT^        TT^        27r^        ilT^        5^^         StT^ 

6'12'4'3'.3'3'2* 

3.  31.416  cm.,  62.832  cm.,  125.664  cm.,  47.124  cm. 

4.  1^,  l\  2^  ^^ 

2  2 

5.  Smaller  sprocket :  4  revolutions  per  second,  angular  velocity  =  8  tt  radi- 
ans per  second,  linear  velocity  of  circumference  =  201.06  inches  per  second; 

167r 


larger  sprocket:  f  revolutions  per  second,  angular  velocity 


3 


radians  per 


second,  linear  velocity  of  circumference  =  201 .06  inches  per  second.     Speed  of 
machine  =  20.944  feet  per  second  =  14.28  miles  per  hour. 


ANSWERS   TO   EXERCISES  143 

6.  Linear  velocity  of  chain  and  of  circumferences  of  both  sprockets  =  75.43 
inches  per  second ;  angular  velocity  of  larger  sprocket  =  15.09  radians  per  sec- 
ond, of  smaller  sprocket  =  37.76  radians  per  second ;  smaller  sprocket  makes 
5.1  revolutions  per  second. 


13. 

I<«<V^- 

14. 

0<«<^,    '{<a 

< 

3  7r          ^^^Stt     37r^^^' 
Exercise  XVI 

4  * 

5. 

(i'l>-    ■ 

-  (!■)■  (V 

,  -1),  etc. 

6. 

(-;,l).etc. 

'  fe  f  )•  { 

5  TT       V2\      , 

Exercise  XVII 

1. 

f. 

7.   120°,  150°,  300°,  330°. 

2. 

l4-2\^ 
3 

8.  15°,  75°,  135°, 

9.  60°,  300°. 

195°,  255°,  315°. 

3. 

5 

10.   30°,  150°,  210< 

^  330°. 

4. 

-If- 

11.   0°,  60°,  180°,  300°,  360°. 

5. 

60°,  300°. 

12.   0°,  150°,  180°, 

210°,  360°. 

6. 

120°,  300°. 

Exercise  XVIII 

1. 

-  sin  20°. 

9. 

0,0. 

22.    0. 

2. 

3. 

-  sin  10°. 
cot  14°. 

10. 

1  +  V3     2V3 
2      '      3    • 

23.  -  .35. 

24.  -1. 

4. 
5. 
6. 

7. 

-  cot  35°. 

-  CSC  30°. 
sec  40°. 

1  +  V3         2V3 
2      '           3    • 

11. 
12. 
19. 

V3 -  1         2V3 

2      '           3    • 
0,0. 
1  +  V3 
4 

25.    +  VI  -  a\ 

26.    ^l--^ 

m 

27.  sin«. 

28.  —  sin  a. 

8. 

1  +  V3         2V3 
2      '          3    • 

20. 
21. 

0. 

0. 

29.  tan  a. 

30.  -  tan  a. 

Exercise  XIX 

5. 
6. 
7. 

a  +  ^  =  sin-i  If  =  cos-i  -  |1, 
cc  +  ft  z=  arcsin  —  ||f  =  arccos 
-sin(a  +  y8).                           9. 

II  quadrant. 
—  Iff,  III  quadrant, 
sin  2  a. 

11.   sin  2  6. 

8. 

cos  (a  +  y8). 

10 

.   cos  2  a. 

12.   cos^. 

13. 

105°- arcsin  ^"  + 
4 

V6 

1                  \/2  -  V6 

=  arccos ; 

4. 

15= 

'  = 

105°      90"  =  arcsin^  7^  = 
4 

\/6+V2 
arccos J 

144  ANSWERS   TO   EXERCISES 

14.   75°  =  arcsm^  +  ^^  =  arccos^-^: 


15°  =  90°  -  75°  =  arcsin^-^  =  arccos^^  +  ^ 


=  arccos- 
4  4 

Exercise  XX 

11.    tan  15°  =  2  -  V3,  cot  15°  ==  2  +  V3. 

15.  sin  (a  +  /8  +  y)  =  sin  a  cos  y8  cos  y  +  cos  a  sin  y8  cos  y  +  cos  a  cos  jS  sin  y 

—  sin  a  sin  y8  sin  y. 

16.  cos  (a+  /3  +  y)  =  cos  ct  cos  ^  cos  y  —  sin  a  sin  y8  cos  y  —  sin  a  cos  ^  sin  y 

—  cos  a  sin  /?  sin  y. 

T  „  ,  .      ,    a   ,      N   tan  a  +  tan  B  +  tan  v  —  tan  «  tan  (3  tan  v 

17.  tan  (a-\-a-\-y)=: ^^ ? 1- ti _L_. 

1  —  tan  ytJ  tan  y  —  tan  y  tan  a  —  tan  a  tan  /3 

18.  cot  (a  +  B+y)=     ^ot^^oty  +  cotycotg+cotctcoty    ^ 

'"^       cot  ct  cot /?  cot  y  —  cot  (/ —  cot /5  —  cot  y 
Note  the  symmetry  in  the  last  four  formulas. 

Exercise  XXI 

5.  2  a  =:  sin-i  yV/o  "=  cos-i  ifif. 

6.  2  a  =  arcsin  ±  ilg  =  arccos  —  ^i|.     Explain  the  signs. 

7.  sin  ^  a  =  ±  f  and  ±  f ,  cos  J  =i  ±  f  and  ±  f. 

a   s4«=.^^^    ana    ±i^:eoa„..l^-    and    .^. 

9.    sin  (a  +  2  ;8)  =  f|f  and  -  ||f,  cos  (a  +  2  ;g;  ..  -^%%  and  -  |04. 
10.    sin  (a-  2  ft)  =±  |f  |  and  T  ff  |,  cos  (a  -  2  ^)  =  T^W  and  T  |tf. 
17.    a  =  30°,  45°,  60°,  210°,  225°,  240°.      18.   a  =  90^  270°,  and  I  arccos  f . 

19.  a  =  67°  30',  157°  30',  247°  30',  337°  30',  and  ^  arctan  f . 

20.  a  =  90°,  270°,    and  i  cos-i  f.        23.   2  x.         24.    1.         25.   0.         26.   1. 

Exercise   XXII 

1.  1  [sin  8  a  +  sin  2  a].  8.   ^  [3  -  4  cos  2  a  +  cos  4  a]. 

2.  ^  [sin  10  a  -  sin  2  a].  9.   -^  [3  sin  2  a  -  sin  6  a]. 

3.  ^  [cos  4  a  -  cos  10  a].  10.   ^^  [1  -  cos  4  a]. 

4.  1  [cos  7  «  +  cos  3  a].  ^5^    ;t .  J;  [/t  =  0,  1,  2,  3,  4]. 

5.  i  [cos  a  -  cos  3  a] .  4  >    >    »    >    j 

6.  K2sin2a  +  sin4a].    •  royfc+n!!:.   p/t  -  0  1   9  31 

7.  i[3  +  4cos2a+cos4a].  ^^'    ^"^  +  ^^4'   L^-«'1'-^J- 

17.  (2^+1)^;   [/j=0,  l,2,...ll]and(3A:+l)|;   [A:  =  0,  1,  2,  3]. 

18.  (2A:  +  1)^;  [^  =  0,  1,2,  ...29]and(2^-+l)^;   [^  =  0,  1,  2,  ...9]. 

19.  A:.|;   [A:  =  0,  1,  2,  ...7].  20.    ^.|;   [^^  =  0,  1,  2,  ...  7]. 


ANSWERS   TO    EXERCISES  145 


Exercise  XXIII 


9.    (2A:  +  1)^;   [/^  -  0,  1,  2,  ...].  13.    ^0^3  a  -  3  cot  oc. 

"^  ^6^  '    '    '      J  3cot2a-l 

kir  ^n     kir  -.  ^     3  tan  a  —  tan^  a 


10.    — .  11.    ^.  14. 

3 

12.    TT  and  (2^  +  1) 


3  4  1-3  tan'-^  a 

15.  4  sin  a  cos^  a  —  4  sin^  a  cos  a. 

16.  8  cos*  «  -  8  cos2  a  -  3. 

17.  0°,  15°,  105°,  180°,  255°,  345^.     (See  Exercise  XIX,  examples  13  and  14.) 

18.  a  =  60°,  90°,  120°,  270°,  and  arcsin ^.  19.    —  •  20.  ^. 

2V3  2  2 

Article  72 
1.    -  sin  0.       2.   sec  0  tan  0.       3.    -  esc  0  cot  ^.       4.    sec^  6.        5.    -  csc^  (9. 

Article  73 

5.  cos^-+  I  sin——;   ±1;  1,  ^ ;   ±1,  ±i;   ±1,  ^ 

_  (2;t  +  l)7r,    .    .     (2^+l)7r      ,    .  .     1±  V^^ 

6.  cos  ^^ — 1^  I  sin  -^^ ^— ;    ±i:   —  1, : 

n  n  2  ' 

±  1  ±«        .    ±  V3±i 

— Ti — ;  ±h  — 7^ 


Article  74 

6.  Products:  15  +  6i;   -5  +  3t;  6  +  12  {;    _8-20^;   12-18/;  4+22i; 

4  +  22  i. 

7.  Quotients:  5-3*;  3  -  2  i.  8.    Results:  4  +  12i;  1;   -  8;  1. 

9.   Roots:  4  -  3  ^  3  +  2  t;   -  46  +  9  i ;  ±21 
10.    ^/a  [cos  ?^  +  i  sin  ?^]  ;  [>l-  =  0,  1,  2,  ...  (n  -  1)]. 


11.    ±1;  ±i;  1,      ^\^    '^  2, -l±V-3. 


INDEX 

[Eeferences  are  to  articles,  except  where  otherwise  indicated.] 


Addition  formulas,  63,  64. 
Ambiguous  case  of  oblique  triangles,  48. 
Angle,  general  definition  of,  52. 
Angles,  positive  and  negative,  3. 
Answers  to  exercises,  page  135. 
Area,  laws  for 

oblique  triangles,  45. 

right  triangles,  17. 

Checks,  20. 

Common  logarithms  of  numbers.  Table  I, 
page  3. 

Complementary  angles,  functions  of,  10. 

Complex  numbers,  graphical  methods  of 
representation  and  combination,  74. 

Composition  of  forces,  51. 

Conversion  formulas  for  products,  69. 

Conversion  formulas  for  sums  and  differ- 
ences, 70. 

Coordinates,  4. 

Definitions  of  the  trigonometric  functions, 

32,  54. 
De  Moivre's  theorem,  73. 
Directed  rectilinear  segments,  2. 
Drawing  instruments,  1. 

Equilibrium  of  forces,  51. 
Errors,  20. 

Exponential  values  of  the  trigonometric 
functions,  75. 

Formulas  for  tan  (a:t0),  cot  (a  i  |3),  66. 
Formulas,  list  of,  page  130. 
Functions  of  0°,  90°,  12. 

of  180°,  36. 

of  270°,  360°,  58. 

of  30°,  45°,  60°,  11. 

of  (90°  + a),  38. 

of  (^~±a],62. 

of  half  an  angle,  68. 
of  twice  an  angle,  67. 
Fundamental  relations  between  the  func- 
tions of  a  single  angle,  9,  34,  59. 


General  inverse  functions,  80. 

Graphs  of    the    trigonometric    functions, 

57. 
Greek  alphabet,  page  x. 

Half  an  angle,  functions  of,  68. 
Hyperbolic  functions,  76. 

Infinity,  definition  of,  12,  35,  58. 

Inverse  functions,  logarithmic  values  of, 

81. 
Inverse  trigonometric  functions,  14. 

Law  for  angles  in  terms  of  sides,  44. 

Law  of  cosines,  42. 

Law  of  projections,  40. 

Laws  of  sines,  41. 

Law  of  tangents,  43. 

Laws  of  area, 

oblique  triangles,  45. 

right  triangles,  17. 
Laws  for  solution  of 

oblique  triangles,  39-44. 

right  triangles,  16. 
Limitations  in  value  of  the  trigonometric 

functions,  8,  33,  55. 
Limits  of  6»/sin  e  and  0/ta.n  e,  72. 
Line  representations  of  the  trigonometric 

functions,  60. 
List  of  formulas,  page  130. 
Logarithms, 

characteristic,  24. 

cologarithms,  28. 

common  system,  23. 

definition  of,  21. 

interpolation,  26. 

laws  of  combination,  22. 

mantissa,  25. 

numbers  from  logarithms,  27. 

of  numbers,  Table  I,  page  3. 

of  trigonometric  functions.   Table  1\ 
page  25. 

Method  of  solution  of  triangles,  18. 
Multiple  angles,  71. 


147 


148 


INDEX 


Natural  trigonometric  functions,  Table 
III,  page  71. 

Oblique  triangles, 

area,  45. 

laws  for  solution,  3f)-44. 
Oblique  triangles,  solution  of,  46-50. 
Orthogonal  projection,  4  (note),  15. 

Periodicity  of  the  trigonometric  functions, 

61. 
Proportional  parts,  theory  of,  79. 
Purpose  of  trigonometry,  5. 

Relation    between    the    ratios    and    the 

angle,  7. 
Resolution  of  forces,  51. 
Right  triangles, 

area  of,  17. 

laws  for  solution,  16. 

solution  by  logarithms,  31. 

solution  by  natural  functions,  18. 

Series,  exponential,  logarithmic,  trigono- 
metric, hyperbolic,  77. 

Signs  of  the  trigonometric  functions,  8,  33, 
55. 

Slide  rule,  30. 

Solution  of  oblique  triangles,  46-50. 


Solution  of  right  triangles, 

by  logarithmic  functions,  31. 

by  natural  functions,  18. 
Squares  of  numbers.  Table  IV,  page  91. 
Subtraction  formulas,  65. 
Supplementary  angles,  functions  of,  37. 

Table  I.   Common  Logarithms   of    Num- 
bers, page  3. 
Table  II.  Logarithms  of  the  Trigonometric 

Functions,  page  25. 
Table  III.   Natural  Trigonometric  Func- 
tions, pagre  71. 
Table  IV.  Squares  of  Numbers,  page  91. 
Trigonometric  functions,  definitions  of, 

for  acute  angles,  6. 

for  obtuse  angles,  32. 

for  the  general  angle,  54. 

logarithms  of,  Table  II,  page  25. 

natural,  Table  III,  page  71. 
Trigonometric  tables,  pages  1-93. 

computation  of,  78. 

description  of,  19. 
Trigonometry,  purpose  of,  5. 
Twice  an  angle,  functions  of,  67. 

Variations  of  the  trigonometric  functions, 
13,  35,  56. 


TABLES 


-/ 


(- 


TABLE    I 


COMMON   LOGARITHMS 

OF    NUMBERS 


N. 

0 

1 
2 
3 

4 
5 
6 

7 
8 
9 

10 

11 
12 
13 

14 
15 
16 

17 
18 
19 

20 

21 
22 
23 

24 
25 
26 

27 

28 
29 

30 


Lo^. 


Infinity. 


o.oo  ooo 
0.30  103 

0.47  712 

0.60  206 
0.69  897 

0.77  815 

0.84  510 
0.90  309 
0.95424 


1.04  139 

1.07  918 
I. II  394 

1. 14  613 
1. 17  609 
1.20  412 

1.23045 
1.25  527 
1.27875 


1.30  103 


I .32  222 
I  34  242 
1  36  173 

1 . 38  02 1 

1.39  794 
1. 41  497 

1.43  136 

1.44  716 
1 .  46  240 


N. 


1.47  712 


30 

31 

32 
33 

34 
35 
36 

37 

38 
39 

40 

41 
42 
43 

44 
45 
46 

47 
48 
49 

50 

51 
52 
53 

54 
55 
56 

57 
58 
59 

60 


Log. 


1.47  712 


.49  136 
.50515 
.51851 

•53  148 
.54  4-'7 
•  55630 

.  56  820 
.  57  978 
.59  106 


60  206 


.61  278 
.62  325 
.63  347 

.64345 
.65  321 
.66  276 

.67  210 
.68  124 
.69  020 


,69  897 


.70757 
.71  600 
.72  428 

.73239 
.74036 

.74819 

•75  587 
76343 
.77085 


1.77  815 


60 

61 
62 
63 

64 
65 
66 

67 
68 
69 

70 

71 
72 
73 

74 
75 
76 

77 

78 
79 

80 

81 

82 
83 

84 
85 
86 

87 
88 
89 


Log. 


1.77  815 


78  533 
.79239 

.79934 

.80618 
.81  291 
.81  954 

.  82  607 
.83251 
.83885 


.84  510 


.85  126 

.85733 
.86332 

.86923 
.87  506 
.88081 

.88649 
.  89  209 
.89763 


90309 


90849 
91  381 

91  908 

92  428 
92942 
93450 

93952 
94448 
94  939 


90  1.95424 


3 


90 

91 
92 
93 

94 
95 
96 

97 
98 
99 

100 

101 
102 
103 

104 
105 
106 

107 
108 
109 

110 

111 
112 
113 

114 
115 
116 

117 
118 
119 


Log. 


1.95424 


1.95  904 

1.96  379 
1 .  96  848 

1.97  313 

1.97  772 

1.98  227 

1.98677 

1 .99  123 
1.99  564 


2 .  00  000 


2.00  432 
2 .  00  860 

2.01  284 

2.01  703 

2.02  119 
2.02  531 

2.02  938 

2.03  342 
2.03743 


2.04139 


2.04  532 

2.04  922 

2.05  308 

2.05  690 
2.06070 
2.06446 

2.06  819 

2.07  188 
2.07  555 


N. 


120 

121 
122 
123 

124 
125 
126 

127 
128 
129 

130 

131 
132 
133 

134 
135 
136 

137 
138 
139 

140 

141 
142 
143 

144 
145 
146 

147 
148 
149 


120  2.07918  160  2.17609 


Log. 


2.07  918 


2.08  279 
2.08  636 

2.08  991 

2.09  342 

2.09  691 

2.10  037 

2.10  380 

2.10  721 

2. 11  059 


2. II  394 


2. 1 1  727 

2.12  057 
2.12  385 

2.12  710 
2.13033 
2.13354 

2.13672 
2.13988 
2.14301 


2. 14  613 


2.14  922 

2.15  229 

2.15  534 

2.15836 

2.16  137 
2.16435 

2.16  732 

2.17  026 
2.17  319 


TABLE  I 


N. 

100 

01 
02 
03 

04 
05 
06 

07 

08 
09 

110 

11 
12 
13 

14 
15 
16 

17 
18 
19 

120 

21 
22 
23 

24 
25 
26 

27 

28 
29 

130 

31 
32 
33 

34 
35 
36 

37 
38 
39 

140 

41 
42 
43 

44 
45 
46 

47 
48 
49 

150 

N. 


O 


oo  ooo 


432 
860 

01  284 

703 

02  119 

531 
938 

03  342 

743 


043 


04  139 


532 
922 

05  308 

690 

06  070 
446 

819 

07  188 
555 


918 


08  279 
636 
991 

09  342 
691 

10  037 

380 
721 

11  059 


394 

727 

12  057 

385 
710 

13  033 

354 

672 

988 

I4_30i^ 

613 


922 

15  229 
534 

836 

16  137 

435 

732 

17  026 

319 
609 


475 
903 
326 

745 
160 

572 

979 
383 
782 


179 


571 
961 
346 

729 
108 
483 

856 

225 
591 


087 


518 

945 
368 

202 
612 

*oi9 

423 
822 


218 


954 


314 
672 

*026 

377 
726 
072 

415 
755 
093 

428 


760 
090 
418 

743 
066 
386 

704 

*oi9 

333 

644 


953 
259 
564 

866 

167 
465 

761 
056 
348 
638 


610 

999 

385 

767 
145 
521 

893 
262 
628 


990 


350 

707 

*o6i 

412 
760 
106 

449 
789 
126 


461 


793 
123 
450 

775 
098 
418 

735 
♦051 

364 


130 


561 
988 
410 

828 

243 

653 

*o6o 

463 
862 


258 


650 

♦038 

423 

805 
183 

558 

930 
298 
664 


*027 


386 

743 
*o96 

447 
795 
140 

483 
823 
160 


494 


826 
156 
483 

808 
130 

450 

767 
*o82 

395 


675  706 

983  *oH 
290 

594 

897 
197 
495 

791 
085 

377 

667 


320 
625 

927 
227 
524 

820 
114 
406 

"696 


73 


604 
030 

452 

870 
284 
694 

*ioo 

503 
902 


297 


689 

*077 

461 

843 
221 

595 

967 

335 
700 

063 


422 

*I32 

482 
830 
175 

517 
857 
193 


528 


860 
189 
516 

840 
162 
481 

799 

*ii4 

426 


737 


'045 
351 
655 

957 
256 

554 
850 
143 
435 
725 


217 


647 
♦072 
494 
912, 
325 
735 

'141 

543 
941 


689 

*ii5 

536 

953 
366 
776 

*i8i 

583 
981 


336 


727 

500 

881 
258 
633 

'004 
372 
737 


^099 


458 

814 

*i67 

517 
864 
209 

551 

890 
227 


561 


893 
222 

548 

872 
194 

513 

830 

*i45 
457 


768 


♦076 
381 
685 

987 
286 
584 

879 
173 

464 

754 


260 


732 

*i57 

578 

995 
407 
816 

*222 
623 
021 


376 


766 

*I54 
538 

918 
296 
670 

*04i 
408 
773 


35 


493 
849 


552 
899 
243 

585 
924 
261 


594 


926 

254 
581 

905 
226 
545 
862 
*I76 
489 


799 


*io6 
412 
715 

*oi7 
316 
613 

909 
202 
493 
782 


303 


415 


805 

*I92 

576 
956 

333 
707 

*o78 

445 
809 


*i7i 


529 
884 
'237 

587 
934 
278 

619 

958 
294 


628 


959 

287 

613 

937 
258 
577 

893 
^208 
520 


829 


*I37 
442 
746 

*o47 
346 
643 

938 

231 
522 

"sTT 


S   9   Prop.  Pts. 


346 


775 

*I99 

620 

♦036 
449 
857 

*262 
663 

*o6o 


454 


817 

♦242 
662 

=078 
490 
898 

*302 

703 

ICX) 

493 


844 

♦231 

614 

994 
371 
744 

=  115 

482 
846 


*207 


565 

920 

'272 

621 

968 

312 

653 

992 
327 


661 


992 

320 

646 
969 

290 
609 

925 

*239 
551 


860 

*i68 

473 

*o77 
376 
673 
967 
260 
551 
840 


389 


883 

♦269 

652 

♦032 
408 
781 

*i5i 
518 
882 


=1=243 
600 
955 

*307 

656 

*oo3 

346 

687 

*02  5 

361 


694 


*024 

352 
678 

*OOI 

322 
640 

956 

*270 

582 
"89? 
♦198 

503 
806 

*io7 
406 
702 

997 
289 
580 


9 


44 

43 

4* 

4-4 

4  3 

4 

8 

8 

8 

6 

8 

13 

2 

12 

9 

12 

17 

6 

17 

2 

16 

22 

0 

21 

5 

21. 

26 

4 

25 

8 

25- 

30 

8 

30 

I 

29 

35 

2 

34 

4 

33 

39 

6 

38 

7 

37- 

41      40     39 


12  o 
16  o 
20  o 

61240 

7  28.0 

8  32.0 
9I36.0 


37 

3-7 
7-4 
II. I 
14-8 
18.5 
22.2 
25  9 
29.6 
33-3 


3  9 

7 
II  7 
15  6 
19  5 
23  4 
27  3 
31-2 
35  » 

36 

3-6 
7- 


35 

34 

33 

3-5 

3  4 

3 

7 

0 

6.8 

6. 

10 

5 

10  2 

9 

14 

0 

136 

13 

17 

5 

17.0 

16. 

21 

0 

20.4 

19 

24 

5 

238 

23 

28 

0 

27  2 

26. 

31 

5 

30.5 

29- 

3a     31     30 


16.0 

19  2 
22.4 
25.6 
28-8 


3-0 

6.C 
9.0 
12  o 


15  5  »50 
18  6  18  o 
21  7 
24  8 
27  9 


21 .0 
24  o 
27.0 


Prop.  Pts. 


LOGARITHMS  OF  NUMBERS 


N. 

O 

1 

2 

3 

4 

5 

6 

y 

S  1  9 

Prop.  Pts. 

150 

51 

17  609 

898 

638 
926 

667 
955 

696 
984 

725 
*oi3 

754 

782 

811 

840 

869 

*04i 

*o7o 

*o99 

*I27 

*i56 

29 

28 

52 

iS  184 

2n 

241 

270 

298 

327 

355 

3^4 

412 

441 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

1 

2 

2.9 

5  8 

2.8 
5.6 

54 

752 

780 

808 

«37 

865 

«93 

921 

949 

977 

*oo5 

3 

8.7 

8.4 

55 

19033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

n.6 

11.2 

56 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

5 

14.5 

14.0 

57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

6 

17.4 

16.8 

58 

893 

921 

948 

976 

*oo3 

♦030 

*o58 

*o85 

*II2 

'/ 

20.3 

19.6 

59 
160 

61 

20  140 

167 

194 

222 

249 

276 

303 

330 
602 

358 
629 

385 
656 

8 
9 

23.2 
26.1 

27 

22.4 
25.2 

26 

412 
683 

439 

466 

493 

520 

548 

575 

710 

737 

763 

790 

817 

844 

871 

898 

925 

62 

952 

978 

*oo5 

*032 

*o59 

*o85 

*II2 

*I39 

*i65 

*,92 

1 

2.7 

5  4 

2.6 
5.2 

63 

21  219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

64 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.1 

7.8 

65 

748 

11^ 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 

10.4 

66 

22  on 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13.5 

13.0 

67 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

6 

7 
8 
9 

16.2 
18.9 
21.6 
24  3 

15.6 
18.2 
20.8 

9.^  A. 

68 

531 

557 

5«3 

608 

634 

660 

686 

712 

737 

.7^3 

69 
170 

71 

789 

814 

840 

866 

891 

917 

943 

,968 

994 

*oi9 

23  045 

070 

096 

121 

147 

172  '  198 

223 

249 

274 

300 

32s 

350 

376 

401 

426 

452 

477 

502 

528 

25 

72 

553 

603 

629 

654 

679 

704 

729 

754 

779 

1 

2  5 

73 

805 

830 

«55 

880 

905 

930 

955 

989 

*oo5 

*o3o 

2 

5.0 

74 

24  055 

080 

los 

130 

155 

180 

204 

229 

254 

279 

3 

7.5 

75 

304 

329 

35^ 

V^ 

403 

428 

452 

477 

502 

527 

4 

10.0 

76 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

5 
6 

7 

12.5 
15.0 

17.5 

77 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*oi8 

78 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

8 

20.0 

79 
180 

81 

285 

527 
768 

310 

334 

358 

3«2 

406 
^648^ 
888~ 

431 
672 
912 

455 

479 

503 

9 

22.5 

551 

575 

600 

624 

864 

696 

720 

744 
983 

.     .  1 

792 

816 

840 

935 

959 

224: 

23 

82 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 

2.4 

2.3 

83 

245 

269 

293 

316 

340 

3^4 

3^7 

411 

435 

458 

2 

4.8 

4.6 

84 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7.2 

6.9 
9.2 

85 

717 

741 

764 

788 

811 

8s4 

858 

881 

90s 

928 

4 

9.6 

86 

951 

975 

998 

*02I 

*o45 

*o68 

*o9i 

Hl^ 

*i38 

*i6i 

5 
6 

12.0 
14  4 

11.5 
13.8 

87 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

7 

16.8 

16.1 

88 

416 

439 

462 

485 

S08 

531 

554 

577 

600 

623 

8 

19.2 

18.4 

89 
190 

91 

646 

669 

692 

715 

738 

761 
989 
217 

784 

807 

830 

852 

9 

21.6 

20.7 

J71 
28  103 

898 
126" 

921 
149 

944 

967 

*OI2 

*o35 

*o58 

*o8i 

..   ..  1 

171 

194 

240 

262 

285 

307 

22  Zi 

21 

92 

330 

353 

375 

398  421 

443 

466 

488 

511 

533 

1 

2.2 

2.1 

93 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2 

94 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

3 
4 

b.b 

8  8 

6.3 
ft  4 

95 

29  003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

;=; 

n  fi 

10.5 
12.6 

96 

226 

248 

270 

292 

314 

336 

35« 

380 

403 

425 

6 

13.2 

97 

447 

469 

491 

513 

535 

557 

579 

601 

623 

64s 

7 

15.4 

14.7 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

8 

17.6 

16.8 

99 
200 

885 
30  103 

907 

929 

951 

973 

994 
211 

*oi6 

*o38 

*o6o 

*o8i 

9 

19.8 

18.9 

125 

146 

168 

190 

233 

255 

276 

298 

0 

1 

2 

3 

4 

5 

6 

7 

«   «| 

Prop.  Pts. 

TABLE  I 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pt8. 

200 

01 
02 
03 

30  103 

320 

535 
750 

12? 

146 

168 

190 

211 

233 

255 

276 

298 

341 
557 
771 

363 
578 
792 

384 
600 

814 

406 
621 
835 

428 

643 
856 

449 
664 
878 

471 
685 

899 

492 
707 
920 

5'1 
728 

042 

1 
2 
3 
4 
5 

22 

2.2 

4.4 

6.6 

8.8 

11.0 

21 

2.1 
4.2 
6.3 
8,4 
10.5 

04 
05 
06 

963 

31  175 

387 

984 
408 

*oo6 
218 
429 

*027 

239 
450 

*048 
260 
471 

*o69 
281 
492 

♦091 
302 
513 

*II2 
323 

534 

♦133 
345 
555 

*I54 
366 
576 

07 
08 
09 

210 

11 
12 
13 

597 
32  015 

618 
827 
035 

639 

848 
056 

"263- 

660 
869 
077 

284 

681 
890 
098 

305 

702 
118 

723 
931 
139 

744 
952 
160 

765 
973 
181 

785 
994 
201 

6 
7 

8 
9 

13.2 
15.4 
17.6 
19.8 

12.6 
14.7 
16.8 
18.9 

222 

243 

325 

346 

366 

387 

408 

'613 
818 

*02I 

428 

634 
838 

449 
654 
858 

469 
675 
879 

490 
695 
899 

510 

715 
919 

531 
736 
940 

756 
960 

572 
777 
980 

593 
797 

*OOI 

1 

20 

2.0 
4.0 

14 
15 
16 

33  041 

244 

445 

062 
264 
465 

082 
284 
486 

102 

304 
506 

122 

526 

143 
546 

163 

365 
566 

183 

385 
586 

203 

224 
425 
626 

3 
4 
5 

6.0 

8.0 

10.0 

17 

18 
19 

220 

21 
22 
23 

646 

846 

34  044 

666 
866 
064 

686 
885 
084 

706 
90s 

104 

726 

925 

124 

746 
945 
143 

766 
965 
163 

786 
985 
183 

806 

*oo5 
203 

826 

*02  5 

223 

6 

7 
8 
9 

1 
2 

12.0 
14.0 
16.0 
18.0 

19 

1.9 

3.8 

242 

439 
635 
830 

262 

282 

301 

321 

341 

361 

380 

400 

420 

616 
811 

*oo5 

459 
850 

479 
674 
869 

498 

518 
908 

537 
733 
928 

557 
753 
947 

577 
772 
967 

596 
986 

24 
25 
26 

35  025 
218 
411 

044 

238 
430 

064 
257 
449 

083 

276 
468 

102 

488 

122 

315 
507 

141 
334 
526 

160 
353 
545 

180 
372 
564 

199 
392 
583- 

3 

4 
5 
6 

7 
8 
9 

5.7 
7.6 
9.5 
11.4 
13.3 
15.2 
17.1 

27 
28 
29 

230 

31 
32 
33 

603 
984 

622 

813 

*oo3 

641 
832 

*02I 

660 

851 

*o4o 

679 

870 

*o59 

698 

889 
*o78 

717 
908 

*o97 

736 

927 

*ii6 

755 
946 

*i35 

774 

965 

*I54 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

..    1 

361 

549 
736 

380 
568 
754 

399 
586 

773 

418 
605 
791 

436 
624 
810 

642 
829 

474 
661 

847 

493 
680 
866 

511 
698 

884 

530 

717 
903 

1 
2 

1» 

1.8 
3.6 

34 
35 
36 

922 

37  107 

291 

940 
125 
310 

959 
144 
328 

977 
162 
346 

996 
181 

365 

*oi4 

If. 

*033 
218 
401 

*o5i 
236 
420 

♦070 
254 
438 

*o88 
273 
457 

3 
4 
5 
6 

5.4 

7.2 

9.0 

10  8 

37 

38 
H9 

240 

41 
42 
43 

475 
658 
840 

493 
676 
858 

1" 
694 

876 

530 
712 
894 

548 

731 
912 

566 
749 
931 

767 
949 

603 
967 

621 
803 
985 

639 

822 

*oo3 

7 

8 
9 

1 
2 

12.6 
14.4 
16.2 

17 

1.7 
3.4 
5.1 

6.8 

8.5 

10.2 

38  021 

039 

057 

075 

093 

112 

292 

471 
650 

130 

148 

166 

184 

202 
561 

220 

399 
578 

238 
417 
596 

256 
614 

274 
632 

489 
668 

328 
507 
686 

346 

525 
703 

364 
543 
721 

44 
45 
46 

739 
39  094 

757 
934 
III 

775 
952 
129 

792 
970 
146 

810 

987 
164 

828 

*oo5 

182 

846 

*023 

199 

863 

^^04 1 

217 

881 

♦058 

235 

899 

♦076 

252 

3 
4 
5 
6 

47 
48 
49 

250 

270 

445 
620 

287 
463 
637 

305 

480 

655 

498 
672 

340 

515 
690 

358 
533 
707 

375 
550 

724 

III 

742 

410 
585 
759 

428 
602 
777 

7 
8 
9 

11.9 
13.6 
15.3 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS 

7 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

250 

51 
52 
52 

39  794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

967 

40  140 

312 

985 
157 
329 

♦002 

175 
346 

♦019 
192 

364 

*037 
209 

381 

*o54 
226 
398 

*o7i 
243 
415 

*o88 
261 
432 

*io6 
278 
449 

*I23 

295 
466 

1 
2 
3 
4 
5 

18 

1.8 
3.6 

5.4 
7.2 
9  0 

54 
55 
56 

483 

654 
824 

500 
671 
841 

518 
688 
858 

535 
705 

875 

552 

722 
892 

569 

739 
909 

586 

756 
926 

603 
773 
943 

620 
790 
960 

637 
807 

976 

57 

58 
59 

260 

61 
62 
63 

993 
41  162 

330 
497 

*OIO 

179 
347 

*027 

196 
363 

*o44 
212 
380 

l47 

*o6i 
229 
397 
564 

*o78 
246 
414 

581 

*o95 
263 
430 

*iii 
280 
447 

*I28 

296 

464 

*i45 
313 
481 

647 

6 

7 
8 
9 

1 
2 

10.8 
12.6 
14.4 
16.2 

17 

1.7 
3.4 

514 

531 

597 

614 

631 

664 
830 
996 

681 
847 

*OI2 

697 
863 

*029 

714 
880 

*o45 

731 

896 

♦062 

747 

913 

*o78 

764 

929 
*o95 

780 

946 

*iii 

797 
963 

*I27 

814 

979 

*i44 

64 
65 
66 

42  160 

325 
488 

177 
341 
504 

193 

357 
521 

210 
374 
537 

226 
390 
553 

406 
570 

259 
423 
586 

275 
439 
602 

292 

455 
619 

308 
472 
635 

3 
4 
0 

5.1 
6.8 
S.5 

67 

68 
69 

270 

71 
72 
73 

651 
813 
975 

667 
830 
991 

684 

846 

♦008 

700 

862 

*024 

716 

878 

*040 

732 

894 

♦056 

217 

749 
911 

*072 

765 

927 

♦088 

781 

943 

*io4 

797 
959 

*I20 

6 

7 
8 
9 

1 
2 

10.2 
11.9 
13.6 
15.3 

10 

1.6 
3.2 

43  136 

152 

169 

185 

201 

233 

249 

265 

281 

297 
457 
616 

313 

473 
632 

329 
489 
648 

345 
505 
664 

361 
521 
680 

377 
537 
696 

393 
553 
712 

409 
569 
727 

425 
584 
743 

441 
600 

759 

74 
75 
76 

775 

933 

44  091 

791 
949 
107 

807 
965 
122 

823 
981 

138 

838 
996 
154 

854 

*OI2 

170 

870 

*028 

185 

886 

*o44 

201 

902 

*059 

217 

917 

*o75 

232 

3 
4 
5 
6 

7 
8 
9 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

14.4 

77 
78 
79 

280 

81 
82 
83 

248 
404 
560 

264 
420 
576 

279 

43^^ 
592 

295 
451 
607 

467 
623 

326 

483 
638 

342 
498 
654 

358 

514 
669 

373 

529 

685 

389 
545 
700 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855_ 

*OIO 

163 
317 

..    1 

871 

45  025 

179 

886 
040 
194 

902 
056 
209 

917 
071 

225 

932 
086 
240 

948 
102 

255 

963 
117 
271 

979 
133 

286 

994 
148 
301 

1 
2 

15 

1.5 
3.0 

84 
85 
86 

332 
484 
637 

347 
500 
652 

362 

515 
667 

378 

530 
682 

393 
545 
697 

408 
561 
712 

423 
576 
728 

439 
591 
743 

454 
606 

758 

469 
621 
773 

3 

4 
5 
6 

4.5 
6.0 
7.5 
9  0 

87 
88 
89 

290 

91 
92 
93 

788 

939 
46  090 

803 

954 
105 

818 
969 
120 

834 
984 
135 

849 
*ooo 

150 

864 

*oi5 

165 

879 

894 

*o45 

195 

909 

*o6o 

210 

924 

*o75 

225 

7 
8 
9 

10.5 
12.0 
13.5 

240 
"389" 

687 

255 

404 

553 
702 

270 

285 

300 

315 

330 

345 

359 

374 

..     1 

419 
568 
716 

434 
583 
731 

449 
598 
746 

464 
613 
761 

479 
627 
776 

494 
642 
790 

509 

657 
805 

672 
820 

1 
2 
3 
4 
5 
6 

J.* 

1.4 
2.8 

94 

95 
96 

835 

982 

47  129 

850 

997 
144 

864 

*OI2 
159 

879 

♦026 

173 

894 

*04i 

188 

909 

*o56 

202 

923 

*o7o 

217 

938 

+085 

232 

953 

*IOO 

246 

967 

*ii4 

261 

4.2 
5.6 
70 

8.4 

97 
98 
99 

300 

276 
422 
567 

290 
436 
582 

305 

451 
596 

319 
465 
611 

334 

480 
625 

770 

349 
494 
640 

784 

363 
509 
654 

799 

378 
524 
669 

I13 

392 
538 
683 

828 

407 

553 
698 

842 

7 
8 
9 

9.8 
11.2 
12.6 

712 

727 

741 

756 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

8 

TABLE  I 

N. 

0 

1 

2 

9 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

300 

01 
02 
03 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

c^57 

48  001 

144 

871 
015 
159 

885 
029 
173 

yoo 
044 
187 

914 
058 
202 

929 

073 
216 

943 
C87 
230 

958 

lOI 

244 

972 
116 
259 

986 
130 
273 

15 

04 
Go 
OG 

287 

430 
572 

302 

444 
586 

316 
458 
601 

330 

473 
615 

344 
487 
629 

359 
643 

373 
657 

387 

401 

544 
686 

416 
558 
700 

1 
2 
3 

1.5 
3.0 
4.5 

07 
08 
09 

310 

11 
12 
13 

996 

728 
869 

*0I0 

742 
883 

*024 

756 

897 

*o38 

770 

911 

♦052 

785 

926 

*o66 

799 

940 

*o8o 

813 
954 

*094 

827 

968 
*io8 

841 
982 

*I22 

4 
5 
6 

7 
8 
9 

6.0 

7.5 

9.0 

10.5 

12.0 

13.6 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 
402 

541 
679 

276 
415 
554 

290 
429 
568 

304 

443 
582 

318 
457 
596 

332 
471 
610 

346 
485 
624 

3^ 
638 

374 
513 
651 

388 
527 
665 

14 
15 
16 

693 
831 
969 

707 

721 

859 
996 

734 
872 

*OIO 

748 
886 

*024 

762 

900 

*o37 

776 

914 

♦051 

790 

927 

*o65 

803 

941 

*o79 

817 

955 
♦092 

1 

14 

1.4 

17 

18 
19 

320 

21 
22 
23 

50  106 
243 
379 

515 

120 
256 
393 
529 

133 
270 
406 

284 
420 

161 
297 

433 

174 
311 

447 

188 

325 
461 

202 
338 
474 

215 
352 

229 

365 
501 

2 
3 
4 
5 
6 
7 
8 
9 

2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
11.2 
12.6 

542 

556 

569 

583 

596 

610 

623 

637 

651 
786 
920 

664 
799 
934 

678 
813 
947 

691 
826 
961 

705 
840 
974 

718 

853 
987 

732 
866 

*OOI 

745 

880 

*oi4 

759 
893 

*028 

772 

907 

*04i 

24 
25 
26 

51  055 
188 
322 

068 
202 
335 

081 
215 
348 

095 
228 
362 

108 
242 
375 

121 

255 
388 

135 

268 
402 

148 
282 
415 

162. 

255 
428 

175 

308 

441- 

27 
28 
29 

830 

31 
32 
33 

455 
587 
720 

468 
601 
733 

481 
614 
746 

495 
627 

759 

508 
640 

772 

654 
786 

534 
667 

799 

548 
680 
812 

693 
825 

574 
706 
838 

1 
2 
3 

4 
5 
6 

7 

13 

1.3 
2.6 
3.9 

5.2 
6.5 
7.8 
9.1 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

983 

52  114 

244 

996 
127 
257 

*oo9 
140 
270 

*022 

III 

*o35 
166 
297 

*048 
179 
310 

*o6i 
192 
323 

*o75 
205 
336 

*o88 
218 
349 

*IOI 

231 
362 

34 
35 
36 

375 
504 

634 

388 

517 
647 

401 

530 
660 

414 

543 
673 

427 
lit 

440 
569 
699 

453 
582 
711 

466 

595 

724 

479 
608 

737 

492 
621 
750 

8 
9 

10.4 
11.7 

37 

38 
39 

340 

41 
42 
43 

763 

892 

53  020 

148 

776 
905 
033 

789 
917 
046 

802 
930 
058 

815 
943 
071 

827 
956 
084 

840 
969 
097 

853 
982 
no 

866 

994 
122 

879 
*oo7 

135 

1 
2 
3 
4 
5 

12 

1.2 
2.4 
3.6 
4.8 
6  0 

161 

173 

186 

199 

212 

224 

237 

250 

263 

275 
403 
529 

288 

415 
542 

301 
428 
555 

314 
441 

567 

326 

453 
580 

339 
466 

593 

352 

479 
605 

364 
491 
618 

377 
504 
631 

390 
643 

44 
45 
46 

656 
782 
908 

668 

794 
920 

681 
807 
933 

694 
820 
945 

706 
832 
958 

719 

845 
970 

732 

7s 

744 
870 

995 

757 

882 

*cx)8 

769 
895 

*020 

6 

7 
8 

7.2 
8.4 
9.6 

47 
48 
49 

350 

54  033 
158 
283 

045 
170 

295 

058 
183 
307 

070 

195 
320 

083 
208 
332 

095 
220 
345 

108 
233 

357 

120 

245 
370 

i 

145 

270 

394 

9 

10.8 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

s 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS 

9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pts. 

350 

61 
52 
53 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

531 
654 
777 

543 
667 
790 

555 
679 
802 

568 
691 
814 

580 
704 
827 

716 
839 

605 
728 
851 

617 

741 
864 

630 

753 
876 

642 
765 
888 

13 

54 
55 
56 

900 

55  023 

145 

913 
035 

157 

925 
047 
169 

937 
060 
182 

949 
072 
194 

962 
084 
206 

974 
096 
218 

986 
108 
230 

998 
121 
242 

*oii 
133 
255 

1 
2 
3 

1.3 
2.6 
3.9 

57 
58 
59 

360 

61 
62 
63 

267 
388 
509 

279 
400 
522 

291 
413 
534 

303 
425 
546 

315 
437 
558 

328 

449 
570 

691 

340 
461 
582 

352 

473 
594 

364 

376 
497 
618 

4 
5 
6 
7 
8 
9 

5.2 
6.5 
7.8 
9.1 
10.4 
11.7 

630 

871 
991 

642 

654 

666 

678 

703 

715 

727 

739 

763 

883 

*oo3 

775 

895 

*oi5 

787 
907 

*027 

799 

919 

*o38 

811 

931 
*o5o 

823 

943 
*o62 

835 

955 

*o74 

847 

967 

♦086 

859 

979 
♦098 

64 
65 
66 

56  no 
229 
348 

122 
241 
360 

134 
253 
372 

146 
265 
384 

158 

277 

.396 

170 
289 
407 

182 
301 
419 

194 
312 
431 

205 
324 
443 

217 
336 
455 

1 

12 

1.2 

67 
68 
69 

370 

71 
72 
73 

467 

585 
703 

478 
597 
714 

490 
608 
726 

502 
620 
738 

5H 
632 
750 

526 

644 
761 

656 
773 

549 
667 

785 

561 
679 
797 

573 
691 
808 

2 
3 
4 
5 
6 
7 
8 
9 

2.4 
3.6 
4.8 
6.0 
7.2 
8.4 
9.6 
10.8 

820 

832 

844 

855 

867 

879 

891 

902 

914 

+031 

148 

264 

926 

*043 

159 
276 

937 

57  054 

171 

949 
066 
183 

961 
078 
194 

972 
089 
206 

984 

lOI 

217 

996 

113 
229 

*oo8 
124 
241 

*oi9 
136 
252 

74 
75 
76 

287 
403 
519 

299 

415 
530 

310 

426 

542 

322 
438 
553 

334 
449 
565 

461 
576 

357 
473 
588 

368 

484 
600 

380 
496 
611 

392 
507 
623 

77 
78 
79 

380 

81 
82 
83 

634 

749 
864 

978 

646 
761 
875 

657 
772 
887 

669 

784 
898 

680 

795 
910 

692 
807 
921 

703 
818 

933 

830 
944 

726 
841 
955 

852 

967 

*o8i 

195 
309 
422 

1 
2 
3 
4 

5 
6 
7 

11 

1.1 
2.2 
3.3 
4.4 
5.5 
6.6 
7.7 

990 

*OOI 

*oi3 

*024 

*o35 

*047 

♦058 

♦070 

58  092 
206 
320 

104 
218 
331 

115 
229 

343 

127 
240 
354 

138 

252 

365 

149 
26^ 
377 

161 

274 
388 

172 
286 
399 

184 
297 
410 

84 
85 
86 

433 
546 
659 

444 
557 
670 

456 
569 
681 

467 
580 
692 

478 

591 
704 

490 
602 
715 

501 

614 
726 

512 
625 
737 

524 
636 

749 

647 
760 

8 
9 

8.8 
9.9 

87 
88 
89 

390 

91 
92 
93 

771 

883 

995 

59  106 

782 

894 
*oo6 

T18 

794 

906 

*oi7 

805 
917 

*028 

816 

928 

♦040 

827 

939 
♦051 

838 

950 

*o62 

850 

961 

♦073 

861 

973 
♦084 

872 

984 

♦095 

1 
2 
3 
4 
5 

10 

1.0 
2.0 
3.0 
4.0 
5  0 

129 

140 

151 

162 

173 

184 

195 

207 

318 
428 

539 

218 
329 
439 

229 
340 
450 

240 
461 

362 
472 

262 

373 
483 

273 
384 
494 

284 

395 
506 

7J> 

517 

306 

417 
528 

94 
95 
96 

550 
660 
770 

671 
780 

572 
682 

791 

693 
802 

594 
704 

813 

605 
824 

616 
726 
835 

627 

737 
846 

638 
748 
857 

649 

6 
7 

8 

6.0 
7.0 
8.0 

97 
98 
99 

400 

N. 

60  097 

"206 

890 
?o1 

901 

*OIO 

119 

912 

*02I 
130 

923 

♦032 

141 

934 

*o43 

152 

945 

"054 

163 

956 

*o65 

173 

966 

+076 

184 

977 
*o86 

195 

9 

9.0 

217 

228 

239 

249 

260 

271 

282 

293 

304 

0 

1 

2 

3 

4 

5 

0 

7 

S 

9 

Prop.  Pts. 

10 

TABLE  I 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pta. 

400 

01 
02 
03 

6o  2o6 

217 

228 

239 

249 

260 

271 

282 

293 

304 

3H 
423 

531 

325 
433 
541 

336 

444 
552 

347 
455 
563 

358 
466 

574 

369 
477 
584 

379 
487 
595 

390 
498 
606 

401 
509 
617 

412 
520 
627 

04 
05 
06 

638 
746 
853 

649 

756 
863 

660 

767 
874 

670 
778 
885 

681 
788 
895 

692 

799 
906 

703 
810 

917 

713 

82  F 
927 

724 
83/ 
938 

P5 
842 

949 

11 

07 

08 
09 

410 

11 
12 
13 

959 

61  066 

172 

970 
077 
183 

981 
087 
194 

991 
098 
204 

*002 
109 
215 

*oi3 
119 
225 

♦023 
130 
236 

*034 
140 

247 

*o45 
151 

257 

*o55 
162 
268 

1 
2 
3 
4 
5 
6 
7 
8 

1.1 
2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

384 
490 
595 

395 
500 
606 

405 

511 
616 

416 
521 
627 

426 
532 
637 

437 
542 
648 

448 

553 
658 

458 
563 
669 

469 

574 
679 

479 
584 
690 

14 
15 
16 

700 
805 
909 

711 
815 
920 

721 
826 
930 

731 
836 
941 

742 
847 
951 

752 
857 
962 

763 
868 
972 

773 
878 
982 

784 
888 
993 

794 

899 

*cx)3 

9 

9.9 

17 
18 
19 

420 

21 
22 
23 

62  014 

118 
221 

024 
128 
232 

034 
138 

242 

045 
149 
252 

055 
159 
263 

066 
170 
273 

076 
180 
284 

086 
190 
294 

097 
201 
304 

107 
211 
315 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

428 
634 

439 
542 
644 

449 
655 

459 
665 

469 

572 
675 

480 

583 
685 

490 

593 
696 

500 
603 
706 

613 
716 

521 
624 
726 

1 
2 

1.0 
2.0 

24 
25 
26 

737 
839 
941 

747 
849 
951 

757 
859 
961 

767 
870 
972 

778 
880 
982 

788 
890 
992 

798 
900 

*002 

808 
910 

*OI2 

818 
921 

*022 

829 

931 

*o33 

3 
4 
5 
6 

3.0 
4.0 
5.0 
6  0 

27 
28 
29 

430 

31 
32 
33 

63  043 
144 

246 

053 

155 
256 

063 
165 
266 

073 
175 
276 

083 
185 
286 

094 

195 
296 

104 
205 
306 

114 
215 
317 

124 
225 
327 

134 
236 

337 

7 
8 
9 

7.0 
8.0 
9.0 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

639 
739 

448 
548 
649 

458 
659 

468 
568 
669 

478 

579 
679 

488 
589 
689 

498 

599 
699 

508 
609 
709 

619 

719 

629 
729 

34 
35 
36 

749 
849 

949 

759 
859 
959 

769 
869 
969 

779 
879 
979 

789 
889 
988 

799 
899 

998 

809 
909 

*oo8 

819 
919 

*oi8 

829 

929 

*028 

839 

939 
*038 

9 

37 

38 
39 

440 

41 
42 
43 

64  048 

147 
246 

058 

157 
256 

068 
167 
266 

078 
177 
276 

088 
187 
286 

098 
197 
296 

108 
207 
306 

118 
217 
316 

128 
227 

326 

137 
237 
335 

1 
2 
3 
4 
5 
6 
7 
8 

0.9 
1.8 

2.7 
3.6 
4.5 
5.4 
6  3 
7.2 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

444 
542 
640 

454 
650 

464 
562 
660 

473 
572 
670 

483 
582 
680 

493 
591 
689 

503 
601 
699 

611 
709 

621 

719 

532 
631 
729 

44 
45 
46 

738 
836 

933 

748 
846 

943 

758 
856 

953 

768 
865 
963 

777 
875 
972 

787 
885 
982 

797 
895 
992 

807 

904 

*002 

816 
914 

826 

924 

♦021 

9 

8.1 

47 

48 
49 

450 

65  031 
128 
225 

040 
137 
234 

050 

147 
244 

060 

157 
254 

070 
167 
263 

079 
176 
273 
369 

089 
186 
283 

099 
196 
292 

108 
205 
302 

118 

215 
312 

408 

321 

331 

341 

350 

360 

379 

389 

398 

N. 

0 

i 

2 

3 

4 

5 

6 

7 

§ 

0 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS 


11 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

450 

51 
52 
53 

65  321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

418 

514 
610 

427 

523 
619 

437 
533 
629 

447 
543 
639 

456 
552 
648 

466 
562 
658 

475 
571 
667 

485 
581 
677 

495 

504 
600 
696 

54 
55 
56 

706 
801 
896 

811 
906 

725 
820 
916 

734 
830 

925 

744 
839 
935 

753 
849 
944 

763 
858 
954 

772 
868 
963 

782 
877 

973 

792 
887 
982 

10 

57 

58 
59 

4G0 

61 
62 
63 

992 

66  087 

181 

276 

*OOI 

096 

191 

*OII 

106 
200 

*020 

115 
210 

*o3o 
124 
219 

*039 
134 
229 

*o49 

143 
238 

♦058 

153 
247 

*o68 
162 
257 

*o77 
172 
266 

1 
2 
3 
4 
5 
6 
7 
8 

1.0 
2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 

285 

295 

304 

3H 

408 
502 
596 

323 

332 

342 

351 

361 

455 
549 
642 

370 
464 
558 

380 

474 
567 

389 
483 

577 

398 
492 
586 

417 

427 
521 
614 

436 

530 
624 

445 
633 

64 
65 
66 

652 

745 
839 

661 

755 
848 

671 
764 

857 

680 

773 
867 

689 

783 
876 

699 

792 
885 

708 
801 
894 

717 
811 
904 

727 
820 

913 

736 
829 
922 

9 

9.0 

67 

68 
69 

470 

71 
72 
73 

932 

67  025 

117 

941 
034 
127 

950 

043 
136 

960 
052 
145 

969 
062 
154 

978 
071 
164 

987 
080 
173 

997 
089 
182 

*oo6 
099 
191 

201 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

1 
2 
3 
4 
6 
6 

9 

0.9 

1.8 
2.7 
3.6 
4.5 
5.4 

302 
486- 

311 
403 
495 

321 
413 
504 

330 
422 

514 

339 
431 
523 

348 
440 
532 

357 
449 
541 

367 
459 
550 

376 
468 
560 

385 
477 
569 

74 
75 
76 

578 
669 
761 

587 
679 
770 

596 
688 
779 

605 
697 
78S 

614 
706 

797 

624 

715 
806 

633 
724 

815 

642 
733 
825 

651 
742 
834 

660 
752 
843 

77 
78 
79 

480 

81 
82 
83 

852 
68  034 

861 
952 
043 

870 
961 
052 

879 
970 
061 

888 

979 
070 

897 
988 
079 

906 

997 
088 

916 

*oo6 

097 

925 

*oi5 

106 

934 

*024 

115 

7 
8 
9 

6.3 
7.2 
8.1 

124 
215 
305 
395 

133 

142 

151 

160 

169 

178 

187 

196 

205 

224 

314 

404 

233 
323 
413 

242 
332 
422 

251 
341 
431 

260 

350 
440 

269 
359 
449 

278 
368 
458 

287 
377 
467 

296 
386 
476 

84 
85 

86 

485 
574 
664 

673 

502 
592 
681 

511 
601 
690 

520 
610 
699 

529 

538 
628 
717 

547 
637 
726 

Pi 
646 

735 

655 

744 

8 

87 
88 
89 

490 

91 
92 
93 

753 
842 

931 

762 
851 
940 

771 
860 
949 

780 
869 
958 

878 
966 

797 
886 

975 

806 
895 
984 

815 
904 
993 

824 
913 

*002 

833 

922 

*9ii 

1 
2 
3 
4 

5 
6 
7 
8 

0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

108 
197 
285 

117 

205 

294 

126 
214 
302 

135 
223 

311 

144 
232 
320 

152 
241 
329 

161 
249 
338 

170 
258 
346 

179 

267 

355 

188 

276 
364 

94 
95 
96 

373 
461 

548 

469 
557 

390 

478 
566 

399 
487 
574 

408 

417 
504 

592 

425 
513 
601 

434 
522 
609 

443 

452 
627 

9 

7.2 

97 

98 
99 

500 

636 

723 
810 

644 
732 
819 

653 

740 
827 

662 

749 
836 

671 
758 
845 
932 

679 
767 
854 

688 

775 
862 

697 
784 
871 

705 
III 

714 
801 
888 

897 

906 

914 

923 

940 

949 

958 

966 

975 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

12 


TABLE  I 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

600 

01 
02 
03 

69  897 

984 

70  070 
157 

906 

914 

923 

932 

940 

949 

958 

966 

975 

992 
079 
165 

*OOI 

088 
174 

*OIO 

096 
183 

*oi8 
105 
191 

*027 

114 

200 

♦036 
122 
209 

*044 
131 
217 

*o53 
140 
226 

*o62 
148 
234 

04 
05 
06 

243 
329 

415 

252 
338 
424 

260 
346 
432 

269 

355 
441 

278 
364 
449 

286 

458 

295 
467 

303 
389 

475 

312 

321 
406 
492 

9 

07 
08 
09 

510 

11 
12 
13 

501 
586 
672 

757 

842 

927 

71  012 

509 

595 
680 

518 
603 
689 

612 
697 

P5 
621 

706 
791 
876 
961 
046 

544 
629 

714 

800 

552 
638 

723 

808 

561 
646 
731 

569 

655 
740 

578 
663 
749 

1 
2 
3 
4 
5 
3 
7 
8 

0.9 
1.8 

2.7 
3  6 
4.5 
0.4 
6.3 
7.2 

766 
851 

935 
020 

774 

859 

944 
029 

783 
868 
952 
037 

817 

825 

834 

885 
969 
054 

893 
978 
063 

902 
986 
071 

910 

995 
079 

919 

*oo3 

088 

U 
15 
16 

265 

105 
189 
273 

"3 

282 

122 

206 
290 

130 
214 
299 

139 
223 

307 

147 
231 

315 

i9'5 
240 

324 

248 

332 

172 
257 
341 

9 

8.1 

17 

18 
19 

520 

21 
22 
23 

349 
433 
517 

357 
441 

525 

366 
450 
533 

374 
458 
542 

383 
466 

550 

391 
475 
559 

399 

483 

567 

408 

492 

575 

416 
500 
584 

592 

8 

10.8 
21.6 

600 

684 
767 
850 

609 

617 

625 

634 

642 

650 

659 

667 

675 

692 

775 
858 

700 
784 
867 

709 
792 
875 

717 
800 
883 

725 
809 
892 

734 
817 
900 

742 
825 
908 

750 

834 
917 

759 
842 
925 

24 
25 

26 

933 

72  016 

099 

941 
024 
107 

950 
032 
115 

958 
041 
123 

966 
049 
132 

975 
057 
140 

983 
066 
148 

991 
074 
156 

999 
082 

165 

*oo8 
090 

173 

0 
4 
5 
6 

3.2 
4.0 

4.8 

27 

28 
29 

530 

31 
32 
33 

181 
263 
346 
428 

189 
272 
354 

198 
280 
362 

206 
288 
370 

214 
296 
378 

222 
304 
387 

230 
313 
395 

239 
321 

403 

247 
329 
411 

255 
337 
419 

7 
8 
9 

5.6 
6.4 
7.2 

436 

444 

452 

460 

469 

477 

485 

493 

501 

509 
591 
673 

518 

599 
681 

607 
689 

534 
616 

697 

1^^ 
624 

705 

550 
632 

713 

558 
640 
722 

567 
648 

730 

575 
656 

738 

583 
665 
746 

34 
35 
36 

^54 
835 
916 

762 

843 
925 

770 
852 
933 

779 
860 
941 

787 
868 

949 

795 
876 

957 

803 
884 
965 

811 

892 
973 

819 
900 
981 

827 
908 
989 

7 

37 

38 
39 

540 

41 
42 
43 

997 

73  078 

159 

239 

*oo6 
086 
107 

*oi4 
094 
175 

*022 
102 
183 

♦030 
III 
191 

*o38 
119 
199 

*o46 
127 
207 

*o54 

135 
215 

*o62 

143 
223 

♦070 

151 
231 

1 
2 
3 
4 
5 
6 
7 
8 

0.7 
1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5  6 

247 

255 

263 

272 

280 

288 

296 

304 

312 

320 
400 
480 

328 
408 
488 

336 
416 
496 

344 
424 
504 

352 
432 
512 

360 
440 
520 

368 
448 
528 

376 
456 
536 

384 

464 

544 

392 

472 

552 

44 

45 
46 

560 
719 

568 
648 
727 

576 
656 

735 

584 
664 

743 

592 
672 
751 

600 
679 
759 

608 
687 

767 

616 
695 
775 

624 
703 
783 

632 
711 
791 

9 

6.3 

47 
48 
49 

650 

957 

807 
886 
965 

815 
894 
973 

823 
902 
981 

830 
910 
989 

838 
918 
997 

846 

926 

*cx>5 

854 

933 

*oi3 

862 
941 

*020 

870 

949 

*028 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

N. 

0     1 

2   3 

4 

5 

6 

7 

S 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS 

13 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pt8. 

51 
52 
53 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

115 
194 
273 

123 
202 
280 

131 
210 
288 

139 
218 
296 

147 
225 

304 

155 
233 
312 

162 
241 
320 

170 
249 
327 

178 
257 
335 

186 
265 
343 

54 
55 
56 

351 
429 
507 

359 
437 
515 

367 
445 
523 

374 
453 
531 

382 
4.61 
539 

390 
468 

547 

398 
476 

554 

406 

484 
562 

414 
492 
570 

421 
500 
578 

57 
58 
59 

560 

61 
62 
63 

IS 

741 

593 
671 

749 

601 
679 

757 

609 
687 
764 

617 
695 

772 

624 
702 
780 

632 
710 
788 

640 
718 
796 

648 
726 
803 

656 

733 
811 

889 

819 

827 

834 

842 

850 

858 

865 

873 

881 

896 

974 

75  051 

904 
981 
059 

912 
989 
066 

920 

997 
074 

927 

*oo5 

082 

935 

*OI2 
089 

943 

*020 
097 

950 

*028 

105 

958 

*o35 

113 

966 

*043 

120 

1 
2 

0.8 
1.6 

64 
65 
66 

128 
205 
282 

136 
213 
289 

143 
220 

297 

151 

228 
305 

159 
236 
312 

166 

243 
320 

174 

328 

182 
259 

335 

189 
266 
343 

197 
274 
351 

3 
4 
5 
6 

2.4 
3.2 
4.0 
4.8 

67 

68 
69 

670 

71 
72 
73 

358 
435 
5" 

587 

366 
442 
519 
595 

374 
450 
526 

458 
534 

389 
465 

542 

618 

397 
473 
549 
626 

404 
481 
557 
633 

412 
488 
565 

420 
496 
572 

427 
504 
580 

7 
8 
9 

5.6 
6.4 
7.2 

603 

610 

641 

648 

656 

664 
740 
815 

671 

747 
823 

679 
831 

686 
762 
838 

694 
770 
846 

702 
778 
853 

709 

785 
861 

717 
868 

724 
800 
876 

732 
808 
884 

74 
75 
76 

891 

967 

76  042 

899 

974 
050 

906 
982 
057 

914 
989 
065 

921 

997 
072 

929 

*oo5 

080 

937 

*OI2 

087 

944 

*020 

095 

952 

*027 

103 

959 

*o35 

no 

77 
78 
79 

580 

81 
82 
83 

118 

193 
268 

125 
200 

275 

III 
283 

140 
215 
290 

148 
223 
298 

155 
230 

305 

1% 

313 

170 

245 
320 

178 

253 

328 

185 

260 

335 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

418 
492 
567 

425 
500 

574 

433 
507 
582 

440 

515 
589 

448 
522 

597 

455 
530 
604 

462 

537 
612 

470 

545 
619 

477 

III 

486 
559 
634 

1 
2 

7 

0.7 
1.4 

84 
85 
86 

87 
88 
89 

590 

91 
92 
93 

641 
716 
790 

864 

938 

77  012 

649 
723 
797 

871 

945 
019 

656 
730 
805 

879 
026 

664 

If. 

886 
960 
034 

671 
819 

893 
967 
041 

678 

753 
827 

901 

975 
048 

686 
760 
834 

908 
982 
056 

842 
916 

989 
063 

701 
775 
849 

923 
997 
070 

708 
782 
856 

930 

*oo4 

078 

3 

4 
5 
6 
7 
8 
9 

2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

159 
232 

305 

166 
240 
313 

173 
247 
320 

181 
254 
327 

188 
262 
335 

269 
342 

203 
276 
349 

210 
283 
357 

217 
291 
364 

225 
298 
371 

94 
95 
96 

379 
452 

525 

386 
459 
532 

393 

466 

539 

401 

474 
546 

408 
481 
554 

415 
488 
561 

422 

430 
503 
576 

437 

583 

444 
517 
590 

97 
98 
99 

670 
743 

605 
677 
750 

612 
685 

757 

619 
692 
764 

627 
699 
772 

634 
706 

779 

641 
714 
786 

648 
721 
793 

866 

656 
728 
801 

663 
735 

GOO 

815 

822 

830 

837 

844 

851 

859 

873 

880 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

14 


TABLE  I 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

600 

01 
02 
03 

n  815 
887 
960 

78  032 

822 

830 

837 

844 

851 

859 

866 

873 

880 

895 
967 
039 

902 

974 
046 

909 
981 
053 

916 
988 
061 

924 
996 
068 

931 
*oo3 

075 

938 

*OIO 

082 

945 

*oi7 

089 

952 

*025 

097 

04 
05 
06 

104 
176 
247 

III 
183 

254 

118 
190 

262 

125 

197 
269 

132 
204 
276 

140 
211 
283 

147 
219 
290 

226 
297 

161 
233 
305 

168 

240 

312 

8 

07 

08 
09 

GIO 

11 
12 
13 

319 

390 
462 

326 

398 
469 

333 
405 
476 

340 

412 
483 

347 
419 
490 

355 
426 

497 

362 

433 
504 

369 
440 
512 

376 
447 
519 

383 

455 
526 

1 
2 
3 
4 
6 
6 
7 
8 

0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

604 

675 
746 

611 
682 
753 

618 
689 
760 

625 
696 
767 

633 

704 
774 

640 
711 
781 

647 
718 
789 

654 

725 
796 

661 
732 
803 

668 

739 
810 

14 
15 
16 

817 

888 
958 

824 

895 
965 

831 
902 
972 

838 
909 
979 

845 
916 
986 

852 
923 
993 

859 

930 

*ooo 

866 

937 
*oo7 

873 

944 

*oi4 

880 
951 

*02I 

9 

7.2 

17 

18 
19 

620 

21 
22 
23 

79  029 
099 
169 

036 
106 
176 

043 
113 
183 

050 
120 
190 

057 
127 
197 

064 

134 
204 

274 

071 
141 

211 

078 
148 
218 

085 

155 
225 

092 
162 
232 

239 

246 

253 

260 

267 

281 

288 

295 

302 

309 
379 
449 

316 

386 
456 

323 
393 
463 

330 
400 
470 

337 
407 

477 

344 
414 

484 

351 
421 
491 

358 
428 
498 

365 
435 
505 

372 

442 

511 

1 
2 

7 

0.7 
1.4 

24 
25 
26 

518 
588 
657 

525 
595 
664 

532 
602 
671 

539 
609 
678 

546 
616 
685 

553 
623 
692 

560 
630 
699 

637 
706 

574 
644 

713 

650 
720 

3 
4 
5 

2.1 
2.8 
3.5 

4  9 

27 

28 
29 

630 

31 
32 
33 

727 
796 
865 

734 
803 
872 

810 

879 

748 
817 
886 

754 
824 
893 

761 

831 
900 

768 

837 
906 

775 
844 
913 

782 
851 
920 

789 
858 
927 

7 

8 
9 

4.9 
5.6 
6.3 

934 

941 

948 

955 

962 
030 

168 

969 

975 

982 

989 

996 

80  003 
072 
140 

010 
079 
147 

017 
085 
154 

024 
092 
161 

037 
106 

175 

044 

051 
120 
188 

058 
127 
195 

06s 
134 
202 

34 
35 
36 

209 

277 
346 

216 
284 
353 

223 
291 
359 

298 
366 

236 
305 
373 

243 
312 

380 

250 

387 

257 
325 
393 

264 

332 
400 

271 

339 
407 

6 

37 

38 
39 

640 

41 
42 
43 

414 
482 
550 

421 
489 
557 

428 
496 
564 

434 
502 
570 

441 
509 

577 

448 

584 
652 

455 
523 

659 

462 
530 
598 

468 
536 
604 

475 
543 
611 

1 
2 
3 
4 
5 
6 
7 
8 

0.6 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4  8 

618 

625 

632 

638 

645 

665 

672 

679 

686 

754 
821 

693 

760 
828 

699 
767 
835 

706 
774 
841 

713 
781 
848 

720 
787 
855 

726 

794 
862 

733 
801 
868 

740 
808 
875 

747 
814 
882 

44 
45 
46 

889 

956 

81  023 

895 
963 
030 

902 
969 
037 

909 
976 
043 

916 
983 
050 

922 
990 
057 

929 
996 
064 

936 

*oo3 

070 

943 

*oio 

077 

949 
♦017 

084 

9 

5.4 

47 
48 
49 

650 

090 
158 
224 

097 
164 
231 

104 
171 
238 

III 
178 
245 

117 
184 
251 

124 
258 

131 

198 
265 

137 
204 
271 

144 
211 
278 

218 
285 

291 

298 

305 

3" 

318 

325 

331 

338 

345 

351 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop  Pts. 

LOGAEITHMS  OF  NUMBERS 

15 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

650 

51 
62 
53 

8 1  291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

358 
425 
491 

365 
498 

438 
505 

378 
445 
511 

385 
518 

391 

458 
525 

398 
465 
531 

405 
471 
538 

411 
478 
544 

418 
485 
551 

54 
55 
56 

|5» 

624 
690 

564 
631 
697 

637 
704 

578 
644 
710 

584 
651 
717 

591 
657 
723 

598 
664 

730 

604 
671 
737 

611 
677 
743 

617 
684 
750 

57 
58 
59 

660 

61 

62 
63 

823 
889 

763 
829 

895 

770 
836 
902 

776 
842 
908 

783 
849 
915 

790 
856 
921 

796 
862 
928 

803 
869 
935 

809 

875 
941 

816 
882 
948 

954 

82  020 

086 

151 

961 

968 

974 

981 

987 

994 

*CXXD 

*oo7 

*oi4 

027 
092 
158 

033 
099 
164 

040 
105 
171 

046 
112 
178 

053 
184 

060 
125 
191 

066 
132 
197 

073 
138 
204 

079 

145 
210 

1 
2 

0.7 
1.4 

64 
65 
66 

217 
282 
347 

223 
289 
354 

230 
295 
360 

236 
302 
367 

Itl 
373 

249 

315 
380 

256 
387 

263 
328 

393 

269 

334 
400 

276 
406 

3 
4 
5 
6 

2.1 

2.8 
3.5 
4.2 

67 
68 
69 

670 

71 
72 
73 

413 
478 
543 

419 
484 
549 

426 
491 
556 

432 
497 
562 

439 
504 

569 

445 
510 

575 

452 
582 

458 

Pi 

465 
530 
595 

471 
536 
601 

7 
8 
9 

4.9 
5.6 
6  3 

607 
672 

802 

614 

620 

627 

633 

640 

646 

653 

659 

666 

679 
III 

685 
750 
814 

692 
756 
821 

698 
763 
827 

705 
769 

834 

711 
776 
840 

718 
782 
847 

724 
789 

853 

730 
795 
860 

74 
75 
76 

866 
930 
995 

872 
937 

*OOI 

879 

943 

*cx)8 

885 

950 

*oi4 

892 
956 

*020 

898 
963 

*027 

905 

969 

*033 

911 

975 
*04o 

918 

982 

*o46 

924 

988 

♦052 

77 
78 
79 

680 

81 
82 
83 

83  059 
123 
187 

065 
129 

193 

072 
136 
200 

078 
142 
206 

085 
149 
213 

091 

155 
219 

097 
161 
225 

104 
168 
232 

no 

174 
238 

117 
181 
245 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

315 
378 
442 

321 

385 

448 

327 
391 

455 

334 
398 
461 

340 
404 
467 

347 
410 

474 

353 
417 
480 

359 
423 
487 

366 
429 
493 

372 
436 
499 

1 
2 

0.6 
1.2 

84 

85 
86 

506 
569 
632 

512 
575 
639 

518 
645 

525 
588 
651 

531 

594 
658 

537 
601 
664 

544 
607 
670 

|5° 
613 

677 

556 
620 
683 

563 
626 
689 

3 

4 
5 
6 

1.8 
2.4 
3.0 
3.6 

87 

88 
89 

690 

91 
92 
93 

696 

759 
822 

702 
765 
828 

708 
771 
835 

778 
841 

721 
784 
^47 

727 
790 
853 

734 
797 
860 

740 
803 
866 

746 
809 
872 

879 

7 
8 
9 

4.2 
4.8 
5.4 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

*oo4 

067 

130 

948 

84  on 

073 

954 
017 
080 

960 
023 
086 

967 
029 
092 

973 
036 
098 

979 
042 
105 

985 
048 
III 

992 
055 
117 

061 
123 

94 
95 
96 

198 
261 

142 

205 
267 

148 
211 
273 

155 

28c 

161 

223 

167 
230 

292 

298 

180 
242 
305 

186 
248 
311 

192 
255 
317 

97 

98 
99 

700 

N. 

323 
448 

330 
392 
454 

336 
398 
460 

342 
404 
466 

348 
410 

473 

354 
417 
479 

361 

485 

367 
429 
491 

373 
435 
497 

379 

4.^2 

504 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

16 

TABLE  1 

N. 

O 

1 

« 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pts. 

700 

01 
02 
03 

84  510 
572 

634 

696 

516 

522 

528 

535 

541 

547 

553 

559 

566 

628 
689 
751 

640 
702 

584 
646 
708 

590 
652 
714 

597 
658 
720 

603 
665 
726 

609 
671 
733 

677 
739 

621 
683 
745 

04 
05 
06 

^57 
819 

880 

763 
825 
887 

770 
831 
893 

776 
837 
899 

782 

844 
905 

788 
850 
911 

794 
856 
917 

800 
862 
924 

807 
868 
930 

813 
874 
936 

7 

07 
08 
09 

710 

11 
12 
13 

942 

85003 

065 

948 
009 
071 

954 
016 
077 

960 
022 

083 

967 
028 
089 

973 
034 
095 

979 
040 

lOI 

985 
046 
107 

991 
052 
114 

997 
058 
120 

1 
2 
3 
4 
5 
6 
7 
8 

0.7 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 

126 

132 

138 

144 

150 

156 
217 
278 
339 

163 

169 

175 

181 

248 
309 

193 

254 

315 

199 
260 
321 

Tel 

327 

211 

272 
333 

224 
285 
345 

230 
291 
352 

236 
297 
358 

242 
303 
364 

14 
15 
16 

370 
431 
491 

376 

437 
497 

382 
443 
503 

388 

449 
509 

394 
455 
516 

400 
461 
522 

406 
467 
528 

412 
473 
534 

418 

479 
540 

425 
485 
546 

9 

6.3 

17 

18 
19 

720 

21 
22 
23 

552 
612 

673 

558 
618 
679 

564 
625 
685 

570 
631 
691 

576 

697 

582 
643 
703 

588 
649 
709 

594 
655 
715 

600 
661 
721 

606 
667 
727 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

794 
854 
914 

800 
860 
920 

806 

866 
926 

812 
872 
932 

818 
878 
938 

824 
884 
944 

830 
890 
950 

836 
896 
956 

842 
902 
962 

848 
908 
968 

1 
2 

6 

0.6 
1.2 

24 
25 
26 

974 

86  034 
094 

980 
040 
100 

986 
046 
106 

992 
052 
112 

998 
058 
118 

*oo4 
064 
124 

*OIO 

070 
130 

*oi6 
076 
136 

*022 
082 
141 

*028 

088 

147 

3 
4 
5 

1.8 
2.4 
3.0 

27 
28 
29 

730 

31 
32 
33 

153 
213 

273 

332 

159 
219 

279 

338 

165 
225 
285 

171 
231 
291 

177 

237 
297 

183 
243 
303 

189 
249 
308 

195 

255 
3H 

201 
261 
320 

207 

267 
326 

6 
7 
8 
9 

3.0 
4.2 
4.8 
5.4 

344 

350 

356 

362 

368 

374 

380 

386 

445 
504 
564 

392 
451 
510 

390 
457 
516 

404 
463 
522 

410 
469 
528 

415 
475 
534 

421 
481 
540 

427 
487 
546 

433 
493 
552 

439 
499 
558 

34 
35 
36 

570 
629 
688 

576 
635 
694 

581 
641 
700 

587 
646 
705 

593 
652 
711 

658 
717 

605 
664 
723 

611 
670 
729 

617 
676 
735 

623 

682 

741 

5 

37 
38 
39 

740 

41 
42 
43 

747 
806 
864 

923 

753 
812 
870 

759 
817 
876 

764 

823 

882 

770 
829 

888 

776 
894 

782 
841 
900 

788 

847 
906 

794 
853 
911 

800 

859 
917 

1 
2 
3 

t 

6 

6 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 

929 

935 

941 

947 

953 

958 

964 

970 

976 

982 

87  04.0 

099 

988 
046 
105 

994 
052 
III 

058 
116 

*oo5 
064 
122 

*OII 

070 
128 

*oi7 
075 
134 

*023 

081 
140 

*029 

087 
146 

*o35 
093 
151 

44 
45 
46 

'57 
216 

274 

163 
221 
280 

169 
227 
286 

175 
233 
291 

181 
239 
297 

186 
245 
303 

192 
251 
309 

198 
256 
315 

204 
262 
320 

210 
268 
326 

9 

4.5 

47 
48 
49 

750 

332 

390 
448 

396 

454 

344 
402 
460 

408 
466 

355 
413 
471 

361 
419 
477 

367_ 
483 

373 
489 

379 
437 
495 

384 
442 
500 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pts. 

LOGAEITHMS  OF  NUMBERS 


17 


N. 

0 

1 

2 

3 

4 

5 

535 

593 
651 
708 

6 

7 

8 

9 

Prop.  Pts. 

750 

51 
52 
53 

87  506 

512 

518 

523 

529 

541 

547 

552 

558 

564 
622 
679 

570 
628 
685 

633 
691 

581 
639 
697 

587 
645 
703 

599 
656 

714 

604 
662 
720 

610 
668 
726 

616 
674 
731 

54 
55 
56 

737 
795 
852 

800 
858 

806 
864 

812 
869 

760 
818 
875 

766 
823 
881 

772 
829 
887 

777 

f35 
892 

783 
841 
898 

789 
846 
904 

57 
58 
59 

760 

61 
62 
63 

910 

967 

88  024 

915 

973 
030 

978 
036 

927 
984 
041 

933 
990 

047 

938 
996 

053 

944 

*OOI 

058 

950 

♦007 

064 

955 

*oi3 

070 

961 

*oi8 

076 

081 

087 

093 

098 

104 

no 

116 

121 

127 

133 
190 
247 
304 

138 
195 
252 

144 
201 
258 

150 
207 
264 

156 

213 
270 

161 
218 

275 

167 
224 
281 

173 

230 

287 

178 

235 
292 

184 
241 
298 

1 
2 

0 

0.6 
1.2 

64 
65 
66 

3?? 
366 

423 

315 

372 
429 

321 
377 
434 

326 

383 
440 

389 
446 

338 
395 
451 

343 
400 

457 

349 
406 

463 

355 
412 
468 

360 
417 
474 

3 
4 
5 

6 

1.8 
2.4 
3.0 
3  6 

67 
68 
69 

770 

71 
72 
73 

480 
536 
593 

485 
542 
598 

491 
604 

497 
553 
610 

502 

|59 
615 

508 
564 
621 

513 
570 
627 

519 

632 

5^5 
581 
638 

F 

643 

7 
8 
9 

4.2 
4.8 
5.4 

649 

655 

660 

666 

672 

(>77 

683 

689 

694 

700 

705 
762 
818 

711 
767 
824 

717 

773 
829 

722 
779 
835 

728 

784 
840 

734 
790 
846 

739 
795 
852 

801 
857 

750 
807 
863 

756 
812 

868 

74 
75 
76 

874 

880 
936 
992 

885 
941 
997 

891 

947 
*oo3 

897 

953 

*oc9 

902 

958 

*oi4 

908 
964 

*020 

913 
969 

*025 

919 

975 
♦031 

925 

98. 

*037 

77 
78 
79 

780 

81 
82 
83 

89  042 
098 
154 

048 
104 
159 

053 
109 
165 

059 

115 
170 

064 
120 
176 

070 
126 
182 

076 
131 

081 
137 
193 

087 

143 
198 

092 
148 
204 

260 

209 
265 

?76 

215 

271 
326 
382 

221 

226 

232 

237 

243 

248 

254 

276 
332 
387 

282 
337 
393 

287 
343 
398 

293 
404 

298 

354 
409 

304 

360 
415 

310 

365 
421 

315 

426 

1 
2 

5 

0.5 
1.0 

84 
85 
86 

542 

437 
492 
548 

443 
498 
553 

448 
504 
559 

454 
509 
564 

459 
515 
570 

465 
520 
575 

470 

526 
581 

476 

We 

481 
537 
592 

3 

4 
5 

p 

1.5 
2.0 
2.5 
3  0 

87 
88 
89 

790 

91 
92 
93 

597 
708 

603 
658 
713 

609 
664 
719 

614 
669 
724 

620 
675 
730 

625 
680 

735 

631 
686 
741 

636 

691 

746 

642 
697 
752 

647 
702 
757 

7 
8 
9 

3.5 
4.0 
4.5 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

818 
873 
927 

823 
878 

933 

829 
883 
938 

834 
889 
944 

840 
894 
949 

845 
900 

955 

851 
905 
960 

856 
911 

966 

862 
916 
971 

867 
922 
977 

94 
95 
96 

982 

90  037 

091 

988 
042 
097 

993 
048 
102 

998 
"III 

*oo4 
059 

i'3 

*oo9 
064 
119 

*oi5 
069 
124 

*020 

075 
129 

*026 

080 
135 

*o3i 
086 
140 

97 
98 
99 

800 

146 
200 

255 

'53 
206 

260 

157 
211 
266 

162 
217 
271 

168 
222 
276 

173 
227 
282 

179 
233 
287 

184 
238 
293 

189 

244 
298 

195 
249 
304 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

18 


TABLE  I 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

800 

01 
02 
03 

90  309 

363 
417 
472 

314 

320 

325 

331 

336 

342 

347 

352 

358 

369 
423 
477 

374 
428 
482 

380 

434 
488 

385 
439 
493 

390 
445 
499 

396 
450 
504 

401 

455 
509 

407 
461 
515 

412 
466 
520 

04 
05 
06 

526 
580 
634 

531 

585 
639 

536 
590 
644 

596 
650 

601 
655 

553 
607 
660 

|5^ 
612 

666 

f3 
617 

671 

569 
623 
677 

574 
628 
682 

07 

08 
09 

810 

11 
12 
13 

687 
741 
795 

693 

747 
800 

698 

752 
806 

703 
811 

709 
763 
816 

714 
768 
822 

875 

720 

773 
827 

881 

725 
779 
832 
886 

730 
784 
838 

-89T 

736 
789 
843 
897 

849 

854 

859 

865 

870 

902 

956 

91  009 

907 
961 
014 

^6^ 

020 

918 
972 
025 

924 

977 
030 

929 
982 
036 

041 

940 

993 

046 

945 
998 
052 

950 

*oo4 

057 

1 
2 

0.6 
1.2 

14 
15 
16 

062 
116 
169 

068 
121 
174 

180 

078 
185 

084 

137 
190 

089 
142 
196 

094 
148 
201 

100 

153 
206 

105 

158 
212 

no 
164 
217 

3 
4 
5 
6 

1.8 
2.4 
3.0 
3  6 

17 
18 
19 

820 

21 
22 
23 

222 

275 
328 

228 
281 
334 

233 
286 

339 

238 
291 

344 

243 
297 
350 

249 
302 

355 

254 
307 
360 

259 
312 
365 

265 
318 
371 

270 

323 

376 

7 
8 
9 

4.2 
4.8 
5.4 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

434 
487 
540 

440 
492 

545 

445 
498 
551 

450 
503 
556 

455 
508 
561 

461 

514 
566 

466 

519 

572 

471 
524 
577 

477 

582 

482 
535 
587 

24 
25 
26 

593 
645 
698 

598 
651 
703 

603 
656 
709 

609 
661 
714 

614 
666 
719 

619 
672 
724 

624 
677 
730 

630 
682 
735 

635 
687 
740 

640 
693 

745 

27 

28 
29 

830 

31 
32 
33 

751 
803 

855 

756 
808 
861 

761 
814 
866 

766 
819 
871 

772 
824 
876 

777 
829 
882 

782 
834 
887 

7^7 
840 
892 

793 
845 
897 

798 
850 

903 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

960 

92  012 

065 

965 
018 
070 

971 
023 
075 

976 
028 
080 

981 

033 

085 

986 
038 
091 

991 
044 
096 

997 
049 

lOI 

*002 
054 
106 

*oo7 
059 
III 

1 
2 

6 

0.5 
1.0 

34 

35 
36 

117 
169 
221 

122 

174 
226 

127 
179 
231 

132 
184 
236 

137 
189 
241 

143 
195 

247 

148 
200 
252 

153 

205 

257 

158 
210 
262 

163 
215 
267 

3 
4 
5 
0 

1.5 
2.0 
2.5 
3  0 

37 
38 
39 

840 

41 
42 
43 

273 
324 
376 

278 
330 
381 

283 
335 
387 

288 
340 
392 

293 
345 
397 

298 

350 
402 

304 
355 

407 

309 
361 
412 

3H 
366 
418 

319 
371 
423 

7 
8 
9 

3.5 
4.0 

4.5 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

480 
531 
583 

485 
536 
588 

490 

542 
593 

495 
547 
598 

500 
603 

505 
557 
609 

562 
614 

516 

567 
619 

521 
572 
624 

526 
578 
629 

44: 

45 

46 

634 
686 

737 

639 
691 
742 

7A7 

650 
701 

752 

655 
706 

758 

660 
711 
763 

665 
716 
768 

670 
722 
773 

675 
727 
77S 

681 
732 
783 

47 

48 
19 

850 
N. 

788 
840 
891 

793 
845 
896 

799 
850 
901 

804 

855 
906 

809 
860 
911 

814 
865, 
916 

819 
870 
921 

824 
875 
927 

829 
881 
932 

834 
886 
937 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  VU. 

LOG 

ARIl 

HMS 

OF  NUMBERS 

19 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

§ 

9 

Prop.  ris. 

850 

51 
52 
53 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

993 

93  044 

095 

998 
049 
100 

*oo3 

054 
105 

*oo8 
059 
no 

*oi3 
064 
115 

*oi8 
069 
120 

*024 

075 
125 

*029 

080 
131 

*oi! 
136 

*039 
090 
141 

54 
55 
5G 

146 
197 

247 

151 

202 
252 

156 
207 
258 

161 
212 
263 

166 
217 
268 

171 
222 
273 

176 

227 

278 

181 

232 

283 

186 

192 
242 
293 

6 

57 

58 
59 

860 

61 
62 
63 

298 
349 
399 

303 
354 
404 

308 

359 
409 

313 
364 
414 

369 
420 

323 
374 
425 

328 

379 
430 

334 
384 
435 

339 

389 
44.0 

344 
394 

445 

1 
2 
3 
4 
5 
6 
7 
8 

0.6 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4.8 

450 
500 

III 

455 

460 

465 

470 

475 
526 
576 
626 

480 

485 

490 

495 

505 
556 
606 

510 
611 

515 
566 
616 

520 

571 
621 

531 

636 

541 
591 
641 

546 
596 
646 

64 
65 
66 

651 
702 
752 

656 
707 
757 

661 
712 
762 

666 
717 
767 

671 
722 
772 

676 
727 
777 

682 
732 
782 

687 
787 

692 
742 
792 

697 
747 
797 

9 

5.4 

67 
68 
69 

870 

71 
72 
73 

802 
852 
902 

807 
857 
907 

812 
862 
912 

817 
867 
917 

822 
872 
922 

827 

877 
927 

832 
882 
932 

837 
887 
937 

842 
892 
942 

847 
897 
947 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

1 
2 
3 
4 
5 
6 

5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 

94  002 
052 

lOI 

007 
106 

012 
062 
III 

017 
067 
116 

022 
072 
121 

027 
077 
126 

131 

086 
136 

042 
091 
141 

047 
096 
146 

74 
75 
76 

151 
201 
250 

156 
206 
255 

161 
211 
260 

166 
216 
265 

171 
221 
270 

176 
226 
275 

181 

186 
285 

191 
240 
290 

196 
245 
295 

77 

78 
79 

880 

81 
82 
83 

300 

349 
399 
448 

305 
354 
404 

310 

359 
409 

364 
414 

320 

369 
419 

325 
374 
424 

330 
379 
429 

335 
384 
433 

340 
438 

345 
394 
443 

7 
8 
9 

3.5 
4.0 
4.5 

453 

458 

463 

468 

473 

478 

483 

488 

493 

498 
547 
596 

503 
552 
601 

507 
557 
606 

512 
611 

567 
616 

522 
621 

527 
576 
626 

532 
630 

537 
586 

635 

r42 

591 
640 

84 
85 
86 

645 
694 
743 

650 
699 
748 

655 
704 

753 

660 
709 
758 

665 
714 
763 

670 
719 
768 

675 
724 

773 

680 

729 
778 

685 
734 
783 

689 
738 
787 

4 

87 
88 
89 

890 

91 
92 
93 

792 
841 
890 

797 
846 
895 

802 
851 
900 

807 
856 
905 

812 
861 
910 

817 
866 

915 

822 

871 
919 

827 
876 

924 

880 
929 

836 
885 
934 

1 
2 
3 
4 
5 
6 
7 
8 

0  4 

0.8 
1.2 
1.6 
2.0 
2.4 
2.8 
3.2 

939 

988 

95  036 

085 

944 

993 
041 
090 

949 

954 

959 

*oo7 

056 

105 

963 

*OI2 
061 
109 

968 

973 

978 

983 

998 
046 
095 

*002 
051 
100 

♦017 
066 
114 

*022 
071 
119 

*027 

075 
124 

♦032 
080 
129 

94 
95 
96 

231 

139 
187 
236 

143 
192 
240 

148 
197 
245 

153 
202 
250 

158 
207 
255 

163 
211 
260 

168 
216 
265 

173 

221 
270 

177 
226 
274 

9 

3.6 

97 

98 
99 

900 

279 

^;6 

284 

332 
381 

289 

337 
386 

294 
342 
390 

299 
347 
395 

303 
352 
400 

308 

357 
405 

361 
410 

318 
366 
415 

323 
371 
419 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

20 

TABLE  I 

■, 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

900 

01 
02 
03 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

472 
569 

477 
525 
574 

482 
530 
578 

487 
535 
583 

492 

540 
588 

497 
545 
593 

501 
598 

506 

554 
602 

511 
559 
607 

564 
612 

04 
05 
06 

617 
665 
713 

622 
670 
718 

626 
674 
722 

631 
679 
727 

636 
684 
732 

641 
6S9 
737 

646 
694 
742 

650 
698 
746 

655 
703 

751 

660 
708 
756 

07 

08 
09 

910 

11 
12 
13 

761 
809 
856 

904 

766 

813 

861 

818 

866 

823 
871 

780 
828 
875 

785 
880 

789 

885 

794 
842 
890 

799 
847 
895 

804 
852 
899 

909 

914 

918 

923 

928 

933 

938 

942 

947 

952 

999 
96  047 

957 

*oo4 

052 

961 

*oo9 

057 

966 

*oi4 
061 

971 

♦019 

066 

976 

*023 

071 

980 

*028 

076 

985 

*o33 

080 

,995 

*042 

090 

1 

2 

0.5 
1.0 

14 
15 
IG 

095 
142 

190 

099 

147 
194 

104 
152 
199 

109 
156 
204 

114 
161 
209 

118 
166 
213 

123 
218 

128 

175 
223 

133 

180 
227 

137 

185 

232 

3 
4 
5 
6 

1.5 
2.0 
2.5 
3.0 

17 

18 
19 

920 

21 
22 
23 

237 
284 
332 

379 

426 

473 
520 

1% 
336 

246 
294 
341 

251 
298 
346 

256 
303 
350 

261 
308 
355 
402 
450 
497 
544 

265 
360 

270 
317 
365 

275 
322 
369 

280 
327 

374 

7 
8 
9 

3.5 
4.0 
4.5 

384 

388 

393 

398 

407 

412 

417 

421 

478 

525 

435 
483 
530 

440 
487 
534 

445 
492 
539 

454 
548 

459 
506 

553 

464 
558 

468 
562 

24 
25 

26 

567 
614 
661 

572 
619 
666 

577 
624 
670 

581 
628 

675 

586 

633 
680 

591 
638 
685 

P5 
642 

689 

600 
647 
694 

605 
652 
699 

609 
656 
703 

27 
28 
29 

930 

31 
32 
33 

708 

755 
802 

848 

713 
759 
806 

717 
764 
811 

722 

^^? 
816 

727 
774 
820 

825 

736 
783 
830 

741 
788 

834 

745 
792 
839 

750 
844 

853 

858 

862 

^7 

872 

876 

881 

886 

890 

895 
942 
988 

900 
946 
993 

904 
951 
997 

909 
956 

*002 

914 
*oo7 

918 
965 

*OII 

923 

970 

*oi6 

928 
974 

*02I 

932 
979 

*025 

♦030 

1 
2 

4 

0.4 

0.8 

34 
35 
36 

97  035 

081 
128 

132 

044 
090 

137 

049 
095 
142 

053 
100 
146 

058 
104 
151 

063 
109 
155 

067 
114 
160 

072 
118 

165 

077 
123 
169 

3 

4 
5 
6 

1.2 
1.6 
2.0 
2.4 

37 
38 
39 

940 

41 
42 
43 

174 
220 
267 

.79 

225 
271 

183 
230 
276 

188 

192 

197 

243 
290 

202 
248 
294 

206 

253 
299 

211 
257 
304 

216 
262 
308 

7 
8 
9 

2.8 
3.2 
3.6 

313 

359 
405 

451 

317 

322 

327 

331 

336 

340 

345 

350 

354 
400 
447 
493 

364 
410 
456 

368 
414 
460 

373 
419 
465 

377 
424 

470 

382 
428 
474 

387 
433 
479 

391 
437 
483 

396 
442 
488 

44 
45 
46 

497 
589 

502 
548 
594 

506 
552 
598 

511 
603 

562 
607 

•520 
566 
612 

525 
571 
617 

529 
575 
621 

534 
580 
626 

630 

47 
48 
49 

950 

635 

681 

727 

640 
685 
731 

644 
690 
736 

649 
695 
740 

653 
699 
745 

658 

704 
749 

663 
708 

754 

667 
713 

759 

672 
717 

763 

676 
722 
768 

772 

m 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

0 

1 

2 

«|4 

5 

6 

7 

8 

9 

Prop.  PtR. 

LOGARITHMS  OF  NUMBERS 

21 

N. 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

050 

51 
52 
53 

97  772 

777 

782  786 

791 

795 

800 

804 

809 

813 

8i8 
864 
909 

823 
868 
914 

827 

873 
918 

832 
^77 
923 

836 
882 
928 

841 
886 
932 

845 
891 

937 

850 
896 
941 

855 
900 
946 

859 
905 
950 

54 
55 
56 

0^55 

98  000 

046 

959 
005 
050 

964 
009 
055 

968 
014 
059 

973 
019 
064 

978 
023 
068 

982 
028 
073 

987 
032 
078 

991 

21 

996 
041 
087 

57 

58 
59 

960 

61 
62 
63 

091 

096 

141 
186 

100 
146 
191 

105 
150 
195 

109 

155 
200 

114 

159 
204 

118 
164 
209 

123 
168 
214 

127 

173 

218 

132 
177 
223 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

272 

363 

277 
322 
367 

281 

327 
372 

286 

331 
376 

290 
336 
381 

295 
340 
385 

299 
345 
390 

304 
349 
394 

308 
354 
399 

313 
358 
403 

1 
2 

5 

0.5 
1.0 

64 
65 
66 

408 

453 
498 

412 

457 
502 

417 
462 

507 

421 
466 
511 

426 

471 
516 

430 
475 
520 

480 
525 

439 
484 
529 

444 
489 

534 

448 

.493 

538 

3 
4 
5 
6 

1.5 
2.0 
2.5 
3.0 

67 

68 
69 

970 

71 
72 
73 

543 
588 
632 

547 
592 
^57 

552 
597 
641 

601 
646 

561 
605 
650 

610 
655 

570 
614 
659 

619 
664 

579 
668 

583 
628 

673 

7 
8 
9 

3.5 
4.0 
4.5 

677 

682 

686 

691 

695 

700 

744 
789 
834 

704 

709 

713 

717 

722 
767 
811 

726 
771 
816 

731 
820 

825 

740 

784 
829 

749 
III 

798 
843 

802 

847 

762 
807 
851 

74 
75 
76 

856 
900 
945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 

923 
967 

883 
927 
972 

^2>7 
932 
976 

892 
936 
981 

896 
941 
985 

77 
78 
79 

080 

81 

82 
83 

989 

99  034 
078 

994 
038 
083 

998 
043 
087 

*oo3 
047 
092 

*oo7 
052 
096 

*OI2 
056 
100 

145 
189 
233 

277 

^=016 
061 
105 

*02I 
065 
109 

*025 

069 
114 

*029 

074 
118 

123 

127 

131 

136 

140 

149 

154 

158 

162 

167 
,  211 

255 

171 
216 
260 

176 
220 
264 

180 

224 
269 

185 
229 

273 

193 

282 

198 

242 
286 

202 

247 
291 

207 
251 
295 

1 

2 

4 

0.4 
0.8 

84 
85 
86 

300 

344 
388 

304 

348 
392 

308 

396 

313 
357 
401 

317 
361 
405 

322 
366 
410 

326 

370 
414 

330 

374 
419 

335 

379 

423 

339 
383 
427 

3 
4 

5 

1.2 
1.6 
2.0 
2  4 

87 
88 
89 

090 

91 
92 
93 

432 
476 
520 

436 
480 

524 

441 
484 
528 

445 
489 
533 

449 
493 
537 

454 
498 
542 

458 
502 
546 

590 

463 
506 

550 

467 
511 
555 

471 
515 
559 
603 

7 
8 
9 

2.8 
3.2 
3.6 

564 

568 

572 

577 

581 

585 
629 
673 
717 

594 

599 

607 

695 

612 
656 
699 

616 
660 
704 

621 
664 
708 

625 
669 
712 

614 

677 
721 

it 

726 

642 
686 
730 

647 
691 
734 

94 
95 
96 

739 
782 
826 

743 
787 
830 

747 
791 

835 

752 
795 
839 

756 
800 
843 

760 
804 
848 

765 
808 
852 

769 

813 
856 

774 
817 
861 

822 
865 

97 

98 
99 

1000 

870 

9'3 
957 

874 
917 
961 

878 
922 
965 

883 
926 
970 

887 
930 
974 

891 

935 
978 

896 
939 
983 

900 
944 
987 

904 
948 
991 

909 
952 
996 

00  coo 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

§ 

9 

Prop.  Pts. 

22 


TABLE  I 


N 

0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  Pts. 

1000 

1001 
1002 
1003 

ooo  ooo 

~434 

868 

ooi  301 

043 

477 
911 

344 

087 

130 

174 

217 

260 

304 

347 

391 

521 

388 

564 
998 
431 

608 

♦041 

474 

651 

♦084 
517 

694 

♦128 

561 

738 

*i7i 

604 

781 

♦214 

647 

824 

♦258 

690 

44 

1004 
1005 
1006 

734 

002  166 

598 

777 
209 
641 

820 

252 
684 

863 
296 
727 

907 

339 
771 

950 
382 
814 

993 

857 

♦036 
468 
900 

*o8o 
512 
943 

*I23 

555 
986 

1 
2 
3 

4.4 

8.8 

13.2 

1007 

1008 
1009 

1010 

1011 
1012 
1013 

003  029 
461 
891 

073 
504 

934 

116 

547 
977 

159 

590 

*020 

202 

633 
*o63 

245 

676 

*io6 

536 

288 
719 

*i49 

331 

762 

*I92 

374 
805 

*235 

417 

848 

*278 

4 
5 
6 
7 
8 
9 

17.6 
22.0 
26.4 
30.8 
35.2 
39.6 

004  321 

364 

407 

450 

493 

579 

622 

665 

708 

751 

005  180 

609 

794 
223 
652 

837 
266 

695 

880 
309 
738 

923 
352 

966 

395 

824 

*oo9 
438 
867 

*052 

481 
909 

*o95 
524 
952 

*i38 
567 
995 

1014 
1015 
1016 

006  038 
466 
894 

081 
509 
936 

124 
552 
979 

166 

594 

*022 

209 

637 
*o65 

252 

680 

*io7 

295 

723 

*i5o 

338 

*i93 

808 
♦236 

f3 
851 

*278 

1 

43 

4.3 

1017 
1018 
1019 

1020 

1021 
1022 
1023 

007  321 
748 

008  174 

600 
oop  026 

451 
S76 

364 
790 
217 

406 

833 
259 

449 
876 
302 

492 
918 

345 

961 
387 

577 
*oo4 

430 

620 

*o46 

472 

662 

*o89 

515 

705 

*I32 

558 

2 
3 
4 
5 
6 
7 
8 
9 

8.6 
12.9 
17.2 
21.5 
25.8 
30.1 
34.4 
38.7 

643 

685 

728 

770 

813 

856 

898 

941 

983 

068 

493 
918 

III 
536 
961 

578 
*oo3 

196 

621 

*o45 

238 

663 

*o88 

281 

706 

*i30 

323 

748 

*i73 

366 
791 

*2I5 

408 

833 
*258 

1024 
1025 
1026 

010  300 

724 

on  147 

342 
766 
190 

232 

427 
851 
274 

470 
893 
317 

936 
359 

978 
401 

597 

*020 

444 

639 

*o63 
486 

681 

*io5 

528 

1027 
1028 
1029 

1030 

1031 
1032 
1033 

570 

993 
012  415 

*035 
458 

655 

''078 

5cx> 

697 

*I20 
542 

740 

*l62 

584 

782 

*204 

626 

824 

*247 
669 

866 

*289 

711 

909 

*33i 

753 

951 

*373 

795 

1 
2 

I 

6 
7 

42 

4.2 

8.4 
12.6 
16.8 
21.0 
25.2 
29.4 

837 

879 

922 

964 

*oo6 

♦048 

♦090 

*I32 

*I74 

*2I7 

013  259 
680 

014  ICX) 

301 

722 
142 

343 
764 
184 

226 

427 
848 
268 

469 
890 
310 

511 
932 
352 

553 
974 
395 

596 

*oi6 

437 

638 

♦058 

479 

1034 
1035 
1036 

521 

940 

015  360 

563 
982 
402 

605 

*024 

444 

647 

*o66 
485 

689 
*io8 

527 

730 

*i5o 
569 

772 

*I92 

611 

814 

*234 

653 

856 

♦276 

695 

898 

*3i8 

737 

8 
9 

33.6 
37.8 

1037 
1038 
1039 

1040 

1041 
1042 
1043 

779 

016  197 
616 

017  033 

821 

239 
657 

863 
281 
699 

904 

323 
741 

946 
365 
783 

988 
407 
824 

+030 

448 

866 

♦072 
490 
908 

*ii4 
532 
950 

♦156 
574 
992 

1 
2 
3 
4 
5 

41 

4.1 

8.2 
12.3 
16.4 
20  5 

075 

117 

159 

200 

242 

284 

326  !  367 

409 

451 

86S 

018  284 

492 
909 
326 

534 
368 

576 

993 
409 

618 

"034 

451 

659 

♦076 

492 

701 

*ii8 

534 

743 
*I59 

576 

784 

*20I 
617 

826 

*243 

659 

1044 
1045 
1046 

700 

019  116 

532 

742 
158 

573 

784 
199 
615 

825 
241 
656 

867 
282 
698 

908 
324 
739 

950 
366 
781 

992 
407 
822 

*o33 
449 
864 

*o75 
490 
905 

6 

7 
8 

24.6 
28.7 
32.8 

1047 

10  i8 
1019 

1050 

947 
020  361 

775 

988 
403 
817 

*030 

444 
858 

*o7i 
486 
900 

*ii3 
527 
941 

*i54 
568 
982 

"3^6 

*I95 
610 

*024 

*237 

651 

*o65 

*278 

693 
*io7 

♦320 

734 
*i48 

561 

9 

36.9 

021  189 

231 

272 

313 

355 

437 

479 

520 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS 

23 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop.  i'U, 

1050 

1051 
1052 
1053 

02I  189 

231 

272 

313 

355 

396 

437 

479 

520 

561 

974 
387 
799 

603 

022  016 

428 

644 
057 

470 

685 
098 
511 

727 
140 
552 

768 
181 
593 

809 

222 
635 

851 
263 
676 

892 

305 
717 

933 
346 
758 

42 

1054 
1055 
1056 

841 

882 

294 
705 

923 

335 
746 

964 
376 
787 

♦005 

417 
828 

*047 
458 
870 

*o88 

499 
911 

*I29 

541 
952 

*i7o 
582 
993 

*2II 
623 

*o34 

1 
2 
3 

4.2 

8.4 

12.6 

1057 
1058 
1059 

1000 

1061 
1062 
1063 

024  075 

486 
896 

025  306 

116 

527 
937 

157 
568 
978 

198 

609 

♦019 

239 

650 

*o6o 

280 
691 

*IOI 

321 
732 

*I42 

363 

773 

*i83 

404 
814 

*224 

445 

855 

♦265 

674 

4 
5 
6 
7 
8 
9 

16.8 
21,0 
25.2 
29.4 
33.6 
37.8 

347 

388 

429 

470 

511 

552 

593 

634 

026  125 

533 

756 
165 
574 

797 
206 
615 

838 
247 
656 

879 
288 
697 

920 
329 
737 

961 
370 
778 

*002 
819 

*o43 
452 
860 

♦084 
492 
901 

1064 
1065 
1066 

942 

027  350 

757 

982 

390 
798 

*023 

431 
839 

*o64 

472 
879 

*io5 

513 
920 

*i46 
961 

♦186 
594 

*002 

*227 

635 
*042 

*268 

676 

*o83 

*309 
716 

*I24 

1 

41 

4.1 

1067 
1068 
1069 

1070 

1071 
1072 
1073 

028  164 
_978 

02Q  384 

789 

030  195 

600 

205 
612 

*oi8 

246 

653 

*o59 

287 
693 

*ICX) 

327 

734 

♦140 

368 

775 
*i8i 

409 

815 

*22I 

449 
856 

*262 

490 

896 

♦303 

531 

937 

*343 

749 

2 
3 

4 
5 
6 

7 
8 
9 

8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
36.9 

424 
830 

640 

465 

871" 
276 
681 

506 

546 

587 

627 

668 

708 

316 
721 

952 

357 
762 

992 
397 
802 

*o33 
438 
843 

*o73 
478 
883 

*ii4 
519 
923 

*i54 
559 
964 

1074 
1075 
1076 

031  004 

408 
812 

045 

449 
853 

085 

489 
893 

126 
530 

933 

166 
570 
974 

206 

610 

*oi4 

247 

651 

*o54 

287 

691 

*o95 

328 

732 

*i35 

368 

772 

*i75 

1077 
1078 
1079 

1080 

1081 
1082 
1083 

032  216 
619 

033  021 

256 
659 
062 

296 
699 
102 

337 
740 
142 

377 
780 
182 

4^7 
820 
223 

458 
860 
263 

498 
901 

J03 

705 

538 
941 

343 

745 

578 
981 
384 
785 

1 
2 
3 
4 
5 
6 
7 

40 

4.0 
8.0 
12.0 
10.0 
20.0 
24.0 
28.0 

424 

464 

504 

544 

585 

625 

665 

826 

034  227 

628 

866 
267 
669 

906 
308 
709 

946 
348 
749 

986 
388 
789 

*027 

428 

829 

*o67 
468 
869 

*io7 
50S 
909 

*I47 
548 
949 

*i87 
588 
989 

1084 
1085 
1086 

035  029 

430 
830 

069 
470 
870 

109 
510 
910 

149 

550 
950 

190 
590 
990 

230 

630 
♦030 

270 

670 

♦070 

310 
710 

*IIO 

350 

750 

*i5o 

390 

790 

♦190 

8 
9 

32.0 
36.0 

1087 
1088 
1089 

1000 

1091 
1092 
1093 

036  230 
629 

037  028 

269 
669 
068 

309 
709 
108 

349 
749 
148 

389 
789 
187 

429 

828 
227 

469 
868 
267 

509 
908 
307 

549 
948 

347 

III 
387 
785 
*i83 
580 
978 

1 
2 
3 
4 

5 

39 

3.9 

7.8 
11.7 
15.6 
19  5 

426 

466 

506 

546 

944 
342 
739 

586 

984 
382 
779 

626 

665 

705 

745 

038  223 
620 

865 
262 
660 

904 
302 
700 

*024 

421 
819 

*o64 
461 

859 

*io3 
501 
898 

*I43 
541 
938 

1094 
1095 
1096 

039  017 

057 

454 
850 

097 

493 
890 

136 

533 
929 

176 
969 

216 

612 

♦009 

?55 
652 

♦048 

295 

692 
*o88 

335 
731 

*I27 

374 

771 

*i67 

6 

7 
8 

23.4 
27.3 
31.2 

1097 
1098 
1099 

1100 

040  207 
602 
998 

246 

642 

*037 

286 

681 

*o77 

325 

721 

*ii6 

761 
♦156 

405 

800 
*i95 

590 

444 
840 

*235 

630 

484 

879 

*274 

669 

523 
919 

*3i4 

708 

563 

958 

"353 

748 

9 

35.1 

041  393 

432 

472 

511 

551 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

§ 

0 

Prop.  Pts. 

TABLE  II 


LOGARITHMS 


OF  THE 


TRIGONOMETRIC  FUNCTIONS 


FOR 


EACH    MINUTE 


25 


26 


TABLE  II 


0^ 


2 

3 
_4_ 

5 
6 

7 
8 

_9_ 
10 
II 

12 

13 

;i 

17 
i8 

19 

20 

21 
22 
23 

24 

26 

27 
28 

29 

30 

31 

32 
33 
34 

36 
37 
38 
39 
40 
41 
42 
43 
44 

II 

49 

60 

51 

52 
53 
il 

ii 
II 

Jl 
00 


L.  Sin. 


6.46373 
6.76476 
6.94085 
7.06  579 


16  270 
24  188 
30882 
36682 
41  797 


46373 
50512 
54291 
57767 
60985 


63982 
66  784 
69417 
71  900 
74248 


76475 
78594 
80615 

82545 
84393 


86  166 
87870 
89509 

91  088 

92  612 


94  084 
95508 
96  §87 
98223 
99520 


8.00779 
8 .  02  002 
8.03  192 
8.04350 
8.05478 


8.06578 
8.07650 
8.08696 

8.09  718 

8.10  717 


II  693 
12647 
13  581 
14495 
15  391 


8.16268 
8.17  128 
8.17971 
8.18798 
8.19  610 


8 .  20  407 
8.21  189 

8.21  958 

8.22  713 
8.23456 


8.24  186 


L.  Cos, 


30103 
17609 
12494 
9691 
7918 
6694 
5800 

5"5 
4576 
4139 
3779 
3476 
3218 
2997 
2802 
2633 
2483 
2348 
2227 

2119 
2021 
1930 
1848 
1773 
1704 
1639 
1579 
1524 
1472 

1424 
1379 
1336 
X297 
1259 
1223 
1190 
1158 
1128 

IIOO 

1072 
1046 

I022 

999 
976 

954 
934 
914 
896 
877 
860 
843 
827 
812 
797 
782 
769 
755 
743 
.730 


L.  Tang,  c.  d.    L,  Cotg 


46373 
76476 
94085 

06579 


16  270 
24  188 
30882 
36682 
41  797 


46373 
50512 
54291 
57767 
60986 


63982 
66785 
69418 
71  900 
74248 


76476 

78595 
80615 
82  546 
84394 


86167 
87871 
89  510 
91  089 
92613 


94  086 
95510 
96889 
98  225 
99522 


00  781 

02  004 

03  194 
04353 
05481 


06581 

07653 

08  700 

09  722 

10  720 


11  696 

12  651 

13585 
14500 
15395 


16  273 

17  133 
17976 
18804 
19  616 


20413 
21  195 

21  964 

22  720 
23462 


8.24  192 


L.  Cotg. 


30103 
17609 

12494 
9691 
7918 
6694 
5800 
5"5 
4576 
4139 
3779 
3476 
3219 
2996 
2803 
2633 
2482 
2348 
2228 
21 19 
2020 
193 1 
1848 
1773 
1704 
1639 
1579 
1524 
1473 
1424 
1379 
1336 
1297 
"59 
1223 
1x90 

"59 
Z128 
ixoo 
1072 
X047 

X022 

998 
976 

955 
934 
915 
895 
878 

860 
843 
828 
812 
797 
782 
769 
756 
742 
730 


c.  d. 


3  53627 
3  23  524 
3  05915 
2.93421 


2.83  730 
2.75  812 
2.69  118 
2.63318 
2  58  203 


2.53627 
2 .  49  488 

2  45  709 
2  42233 
2.39014 


2.36018 

2  33215 
2  30  582 
2  28  100 
2  25  752 


2  23  524 
2  21  405 
2  19385 


7  454 
[5  606 


13833 
12  129 
10  490 
08  911 
07387 


05914 
04  490 
03  III 

01  775 
00478 


99219 
97996 
96806 
95647 
94519 


93419 
92347 
91  300 
90278 
89280 


88304 

87349 
86415 
85  500 
84605 


83727 
82867 
82  024 
81  196 
80384 


79587 
78805 
78036 
77  280 
76538 
75808 


L.  Tang, 

89^ 


L.  Cos. 


000 
000 
000 
000 
000 


000 
000 
000 
000 
oco 


000 
000 
000 
000 
000 


00  000 
00  000 
99  999 
99  999 
99  999 


99  999 
99  999 
99  999 
99  999 
99  999 


99  999 
99  999 
99  999 
99  999 
99998 


99998 
99998 
99998 
09998 
99998 


99998 
99998 
99997 

99  997 
99  997 


99997 
99  997 
99  997 
99997 

99996 


99996 
99996 
99996 
99996 
99996 


99  995 
99  995 
99  995 
99  995 
99  995 


99  994 
99  994 
99  994 
99994 

99  994 


99  993 


L.  Sin. 


35 
34 
33 
32 
3» 
30 
29 
28 
27 
26 


Prop.  Pts. 


3476 

3218 

.X 

348 

322 

.2 

695 

644 

•3 

1043 

965 

•4 

1390 

1287 

•5 

1738 

X609 

2803 

2633 1 

.1 

280 

263 

.2 

560 

527 

•3 

841 

790 

•4 

1121 

1053 

•5 

1401 

X316 

2227 

203I 

.1 

223 

202 

.2 

445 

404 

•3 

668 

606 

•4 

891 

808 

•5 

IXX3 

xoxo 

1704 

1579 

.1 

170 

158 

.2 

341 

316 

•3 

5" 

474 

■4 

682 

632 

•5 

852 

789 

1379 

1297 

.X 

138 

130 

.2 

276 

259 

•3 

414 

389 

•4 

552 

519 

•5 

690 

649 

XX58 

1x00 

X16 

110 

232 

220 

347 

330 

463 

440 

579 

550 

999 

954 

.1 

100 

95 

.2 

200 

191 

•3 

300 

286 

•4 

400 

382 

•5 

500 

477 

877 

843 

.1 

88 

84 

.2 

»75 

169 

•3 

263 

253 

•4 

351 

337 

•5 

438 

422 

782 

755 

.1 

78 

75 

.a 

156 

i5» 

•3 

235 

226 

•  4 

313 

302 

•5 

391 

377 

2997 

300 

599 
899 
1 199 
1498 

2483 

24& 

497 
745 
993 
1242 

1848 

185 
370 
554 
739 
924 

147a 

H7 
294 
442 
589 
736 

1223 
122 
245 
367 
489 
612 

X046 

105 
209 
314 
418 
523 

914 

91 
X83 
274 
366 

457 

8ia 

81 
162 
244 
325 
406 

730 
73 
146 
219 
29a 
365 


Prop.  Pte. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


27 


L.  Sill. 


0 

\ 

2 

3 

_4_ 

I 

7 
8 

_9_ 

10 

12 

13 

il. 

;i 

17 

t8 

20 

21 
22 
23 
24 

26 

27 

28 

29 


31 

32 

33 
34 

36 
37 
38 
ii 
40 
41 
42 
43 
44 

45" 
46 

47 
48 

60 

51 
52 
53 
il. 

^i 
II 

59 
60 


24  186 
24903 

25  609 
26304 
26988 


27  661 

28  324 
28977 

29  621 
30255 


30879 

31  495 

32  103 

32  702 

33  292 


33875 
34450 
35018 

35578 
36  131 


36678 
37217 
37750 
38276 
38796 


39310 
39818 
40320 

40  816 

41  307 


41  792 
42272 

42  746 
43216 

43  680 


44  139 

44  594 

45  044 
45489 
45930 


46  366 

46  799 

47  226 
47650 

48  069 


48485 
48896 

49304 

49  708 

50  108 


50504 
50897 
51  287 
51  673 
52055 


52434 
52  810 
53183 
53552 
53919 


54282 


L.  Cos. 


717 
706 
695 
684 

673 
663 

653 
644 

634 
624 

616 
608 
599 
590 
583 

575 
568 
560 
553 
547 
539 

533 
526 
520 
514 
508 
502 
496 
491 
485 
480 

474 
470 
464 
459 
455 
450 
445 
441 
436 

433 
427 

424 
419 
416 
411 
408 
404 
400 
396 

393 
390 
386 
382 
379 
376 
373 
369 
367 
363 


Tang. 


24  192 

24  910 

25  616 

26  312 
26  996 


27  669 

28  332 
28986 

29  629 
30263 


30888 

31  505 

32  112 

32  711 

33302 


33886 
34461 
35029 
35590 
36143 


36689 
37229 
37762 
38289 
38809 


39323 
39832 
40334 
40830 
41  321 


41  S07 

42  287 
42  762 
43232 
43696 


44  156 

44  611 

45  061 
45507 
45948 


46385 
46817 

47245 
47669 
48089 


48505 
48917 

49325 

49  729 

50  130 


50527 

50  920 

51  310 
51  696 
52079 


52459 
52835 
53208 
53578 
53  945 


54308 


L.  Cotg. 


c.d. 


718 
706 
696 
684 
673 
663 
654 
643 
634 
625 

617 
607 

599 
591 
584 

575 
568 
561 
553 
546 

540 
533 
527 
520 
514 

509 
502 
496 
491 
486 
480 
475 
470 
464 
460 

455 
450 
446 
441 
437 

432 
428 
424 
420 
416 
412 
408 
404 
401 
397 
393 
390 
386 

383 
380 

376 
373 
370 
367 
363 


C.d. 


L.  Cotg. 


75808 
75090 
74384 
73688 
73004 


72331 
71  668 
71  014 
70371 
69  737 


69  112 
68495 
67888 
67  289 
66698 


66  114 

65539 
64971 
64  410 
63857 


63  3" 
62771 
62238 
61  711 
61  191 


60  677 
60168 
59  666 
59  170 
58679 


58193 
57713 
57238 
56768 
56304 


55844 
55389 
54  939 
54  493 
54052 


53615 
53183 
52755 
52331 
51  9" 


51495 
51083 
50675 
50  271 
49870 


49  473 
49  080 
48  690 

48304 
47921 


47541 
47165 
46  792 
46  422 
46055 


.45692 


L.  Tang. 

88° 


L.  Cos. 


99  993 
99  993 
99  993 
99  993 
99992 


99992 
99992 
99992 
99992 
99991 


99991 
99991 
99990 
99990 
99990 


99990 
99989 
99989 
99989 
99989 


99988 
99988 
99988 
99987 
99987 


99987 
99  986 
99  986 
99  986 
99985 


99985 
99985 
99984 
99984 
99984 


99983 
99983 
99983 
99  982 
99982 


99982 
99981 
99981 
99981 
99  980 


99  980 
99  979 
99  979 
99979 

99978 


99978 
99977 

99  977 
99977 
99976 


99976 
99  975 
99  975 
99  974 
99  974 


9  99  974 


L.  Sin, 


GO 

59 
58 
57 
_5i 
55 
54 
53 
52 
_51 
50 
49 
48 
47 
46 


45 
44 
43 
42 

_iL 
40 

39 
38 

_3^ 

35 
34 
33 
32 

30 

29 
28 
27 
26 


25 
24 
23 
22 
21 
20" 

19 
18 

17 
16 


15 
14 
13 
12 
II 

To" 
9 

8 

7 
6 


Prop.  Pte. 


717 

695 

.1 

71.7 

69.5 

.2 

M3-4 

139.0 

•3 

215. 1 

208.5 

•4 

286.8 

278.0 

•  5 

358.5 

347-5 

653 

634 

.1 

65.3 

63-4 

.2 

130.6 

126.8 

•3 

195-9 

190.2 

•4 

261.2 

253.6 

•5 

326.5 

3170 

599 

583 

59-9 

58.3 

119. 8 

n6.6 

179.7 

174.9 

239-6 

233-2 

299-5 

291-5 

553 

539 

.1 

55-3 

53-9 

.2 

no. 6 

107.8 

■3 

165.9 

161.7 

•4 

221.2 

215.6 

•5 

276.5 

269.5 

514 

503 

51.4 

50.2 

102.8 

100.4 

154-2 

150.6 

205.6 

200.8 

257.0 

251.0 

480 

470 

.1 

48 

47 

.2 

96 

94 

•3 

144 

141 

•4 

192 

188 

•5 

240 

235 

450 

440 

.X 

45 

44 

.2 

90 

88 

•3 

135 

132 

•4 

180 

176 

•5 

225 

220 

430 

410 

.1 

42 

4» 

.2 

84 

82 

•3 

126 

123 

•4 

168 

164 

•5 

210 

205 

390 

380 

39 

38 

78 

76 

117 

114 

156 

152 

>95 

190 

673 

67.3 

134.6 

201  9 
269.2 
336.5 

616 

61.6 
123.2 
184.8 
246.4 
308.0 


56.8 
113.6 
170.4 
227.2 
284.0 

536 

52.6 
105.2 
157.8 
210.4 
263.0 

490 

49 

98 

U7 

196 

245 

460 

46 
92 

138 
184 
230 

430 

43 
86 
129 
172 
215 

400 

40 
80 
120 
160 

300 

37 

74 

III 

148 


Prop.  Pts. 


28 


TABLE  II 


2' 


3 

±^ 

l 

I 

9_ 
10 
II 

12 

13 

:i 

ii. 

20 

21 
22 
23 

24 

25 
26 
27 
28 
29 

30 

31 

32 
33 
11 
35 
36 
37 
3« 
39. 
40 

41 

42 

43 
44 

46 

47 
48 

49 

50 

51 
52 
53 

54 

II 

57 
58 
59 
GO 


L.  Sin. 


8  54  282 
8 . 54  642 
8.54999 
8-55  354 


8 .  56  054 
8 .  56  400 

8.56743 
8.57084 
8.57421 


8.57757 
8.58089 
8.58419 

8.58747 
8.59072 


8  59  395 
8.59715 
8 .  60  033 
8.60349 
8.60662 


8.60973 
8.61  282 
8.61  589 

8.61  894 

8.62  196 


8 .  62  49  7 

8.62  795 

8.63  091 
8.6338c 
8.6367 


8.63968 
8.64256 
8.64543 
8.64827 

8.65  no 
8.65391 
8.65670 
8.65947 
8 .  66  223 

8.66  497 


8.66  769 
8  ■  67  039 
8.67.308 

8.67575 
8.67841 


8.68  104 
8.68367 
8.68627 

8.68  886 

8.69  144 


8 .  69  400 
8 .  69  654 

8.69  907 

8.70  159 
8 .  70  409 


8.70658 
8  70  905 
8.71  151 
8.71  395 
8.71638 


8.71  880 


L.  Cos. 


360 
357 

355 
351 
349 
346 
343 
341 
337 
336 

332 
330 
328 

325 
323 
320 
318 
316 
313 
3" 

309 
307 
305 
303 
301 
298 
296 
294 
293 
290 

288 
287 
284 
283 
281 
279 
277 
276 
274 
272 
270 
269 
267 
266 
263 
263 
260 
259 
258 
256 

254 
253 
252 
250 
249 

247 
246 
244 
243 
242 


L.  Tang. 


c.d. 


8.54308 
8 .  54  669 
8.55027 
8.55382 
8  55  734 


8  56~oS3" 

8.56  429 
8.56773 

8.57  "4 
8.57452 


8.57788 
8  58  121 
8  58451 
8  58779 
8.59  105 


8.59428 

8  59  749 
8.60068 
8.60384 
8.60698 


8.61  009 
8.61  319 
8.61  626 

8.61  931 

8.62  234 


8.62535 
8.62834 
8.63  131 
8.63426 
8.63718 


8 .  64  009 
8.64298 
8.64585 
8.64870 
8.65  154 


8.65435 
8.65  715 
8.65993 
8.66269 
8.66543 


8.66816 
8.67087 
8.67356 
8.67624 
8  67890 


8.68  154 
8.68417 
8.68678 
8.68938 

8.69  196 


8.69453 

8.69  708 
8 .  69  962 

8.70  214 
8 .  70  465 


8.70714 
8 .  70  962 
8.71  208 

8.71453 
8.71  697 


8.71  940 


L.  Cotff. 


361 
358 
355 

352 
349 
346 
344 
341 
338 
336 

333 
330 
328 
326 
323 
321 
319 
316 
3H 
3" 
310 
307 
305 
303 
301 

299 
297 
295 
292 
291 
289 
287 
285 
284 
281 
280 
278 
276 
274 
273 
271 
269 
268 
266 
264 
263 
261 
260 
258 
257 
255 
254 
252 
251 
249 

248 
246 
245 
244 
243 


L.  Cotg. 


1.45692 
1-45  331 
1.44  973 
I .44  618 
1 .  44  266 


1-43917 
I -43  571 
1.43227 
1.42886 
1 .  42  548 


I .42  212 
1. 41  879 

I -41  549 
1 .41  221 

1.40895 


1.40572 
1 .40  251 

1-39932 
1.39  616 
1-39302 


•38991 
.38681 

-38374 
.38069 
.37766 


•27  ^§ 
.37  166 

•  36869 

-36574 
.  36  282 


1-35  991 
1.35  702 

1-35415 
1-35  130 
1.34846 


1-34565 
1-34285 
1.34007 
1-33  731 
I  33  457 


I  33  184 
I  32913 
1.32644 
1.32376 
I  32  no 


C.d, 


1. 31  846 

I -31  583 
1. 31  322 
1 .31  062 
I  30  804 


I  30547 
1 .  30  292 
1 .  30  038 
I . 29  786 
1-29  535 


1 .  29  286 
1 .  29  038 
I . 28  792 
1.28547 
1.28303 


1.28060 


L.  Tang. 

87^ 


L.  Cos. 


99  974 
99  973 
99  973 
99972 

99972 


99971 
99971 
99970 
99970 
99969 


99969 
99  968 
99  968 
99967 
99967 


99967 
99  966 
99  966 
99965 
99964 


99964 
99963 
99963 
99  962 
99  962 


99961 

99  961 
99  960 
99  960 
99  959 


99  959 
99958 
99958 
99  957 
99956 


99956 

99  955 

9  99  955 

9  99  954 

9-99  954 


99  953 
99952 
99952 

99951 
99951 


99950 
99  949 
99  949 
99948 
99948 


99  947 
99946 
99946 
99  945 
99  944 


99  944 
99  943 
99942 
99942 
99941 


99940 


L.  Sin. 


GO 

59 
58 

57 

55 
54 
53 
52 
il 
50 
49 
48 
47 


25 
24 

23 
22 
21 
20 

19 
18 

17 
16 


15 
14 
13 
12 
II 

To 

9 

8 

7 
6 


Prop.  Pte. 


360 

350 

36 

35 

72 

70 

108 

105 

144 

140 

180 

175 

216 

210 

•  7 

252 

245 

.8 

288 

280 

•9 

324 

315 

330 

320 

1 

33 

32 

3 

66 

64 

3 

99 

96 

4 

132 

128 

5 

165 

160 

6 

108 

192 

7 

231 

224 

8 

264 

256 

9 

297 

288 

300 

290 

385 

30 

29 

28. 

60 

58 

57 

90 

87 

85- 

120 

116 

114. 

150 

145 

142. 

.6 

180 

174 

171. 

•7 

210 

203 

199. 

.8 

240 

232 

228. 

•9 

270 

261 

256 

380 

375 

270 

I 

28.0 

27-5 

27- 

.2 

56.0 

55-0 

54- 

•3 

84.0 

82.5 

81. 

•4 

112. 0 

IIO.O 

108. 

•  5 

140.0 

137-5 

135- 

.6 

168.0 

165.0 

162. 

-7 

196.0 

192.5 

189. 

.8 

224.0 

220.0 

216. 

•9 

252.0 

247 -5 

243- 

365 

■  26.5 
•53-0 
•79-5 
106.0 
132-S 
159.0 
1.85  5 
212  o 


260 

.26.0 
.52.0 
.78.0 
104.0 
130.0 
156.0 
182.0 
208.0 
234.0 


250 

245 

.25.0 

•24.5 

.50 

0 

-49 

0 

•75 

0 

•73 

5 

100 

0 

198 

0 

125 

0 

122 

5 

.6 

150 

0 

»47 

0 

'75 

0 

171 

5 

200 

0 

196 

0 

•9 

225 

0 

320 

5 

Prof).  Pts. 


LOGARITHMS  OP  THE  TRIGONOMETRIC  FUNCTIONS         29 


3° 


2 

3 
j4 

I 

7 
8 

9 
10 

II 

12 

13 

J4_ 

;i 
\i 

Ji_ 

20 

21 

22 
23 
24 

26 

27 
28 

29 

30 

31 

32 

33 

36 

37 
38 

40 

41 
42 
43 
44 

45 
46 
47 
48 

19. 
50 

51 

52 
53 
54. 
55 
56 

II 

59 
(JO 


L.  Sin. 


8  74  226 

8.74454 
8.74680 
8 .  74  906 
■  75  130 


71  880 

72  120 
72359 
72597 
72834 


73069 
73303 
73  535 
73767 
73  997 


75  353 
75  575 
75  795 
76015 
76234 


76451 

76  667 
76883 
77097 
77310 
77522 

77  733 
77  943 
78152 

78360 


78568 

78774 
78979 

79183 
79386 


79588 
79789 
79990 
80189 
80388 


80585 
80782 
80978 
81  173 
81367 


8.B1  560 
8.81  752 

8.81  944 

8.82  134 
8.82324 


82513 
82  701 
82888 

83075 
83261 


83446 
83630 
83813 
83996 
84177 


8.84358 


L.  Cos. 


240 

239 
238 

237 
23s 
234 
232 
233 
230 
229 
228 
226 
226 
224 
223 
222 
220 
220 
219 
217 
216 
216 
214 
213 
212 

211 
210 

209 
208 
208 
206 
205 
204 
203 

20-i. 
201 
201 
199 
199 
197 
197 
196 

194 
193 
192 
192 
190 
190 
189 
188 
187 
187 
186 
185 
184 
183 
183 
181 


L.  Tang. 


c.  d. 


71  940 

72  181 
72  420 
72659 
72  896 


73366 
73  600 
73832 
74063 


74292 
74521 
74748 

74  974 

75  199 


75423 
75645 
75867 
76087 
76306 


76525 
76742 
76958 
77173 
77387 


77  600 

77  811 

78  022 
78232 
78441 


78649 

78855 

79  061 

8  79  266 

8.79470 


79673 
79875 
80076 
80  277 
80476 


80  674 
80872 

81  068 
81  264 
81  459 


8.81  653 
8  81  846 
8  82038 
8  82  230 
8  82  420 


8  82610 
8  82  799 
8  82987 

883175 
8.83361 


8  83547 
8  83732 
8  83  916 
8  84  100 
8  84282 


8,84464 


L.  Cotg. 


241 

239 
239 
237 
236 

234 
234 
232 
231 
229 
229 
227 
226 
225 
224 
222 
222 
220 
219 
219 

217 
216 
215 
214 
213 
211 


209 
208 
206 
206 
205 
204 
203 
202 
201 

20I 
199 

198 
I9& 
196 
196 

194 

193 
192 
192 
190 
190 
189 


186 
186 

185 
184 
184 
182 


L.  Cotg. 


1 .  28  060 
1.27  819 

1.27  580 
I  27341 
I .27  104 


1.26868 
1.26634 
1 .  26  400 
1.26  168 
1-25937 


I . 25  708 

I  25  479 
I .25  252 
I .25  026 
1 .  24  801 


I  24577 

1-24355 
I  24  133 
1-23913 
1.23694 


I  23  475 
1.23258 
1.23042 
1.22  827 
I .22  613 


1 .  22  400 
I .22  189 
1. 21  978 
1. 21  768 
I-21  559 


I  21  351 
I  21  145 
1.20939 
1.20734 
1.20530 


c.  d. 


1 .  20  327 
I .20  125 
I  19  924 
I  19  723 
I  19  524 
1. 19  326 
1. 19  128 
1 .  18  932 

1. 18  736 

1. 18  541 


18347 
18  154 
17  962 
17770 
17580 


1. 17 390 
1.17  201 
1.17013 
1. 16  825 
I  16  639 


I  16453 
1 .  16  268 
1 .  16  084 
1 .  1 5  900 
1.15  718 


15  536 


L.  Tang. 

80° 


L.  Cos. 

9  99940 
9  99940 
9  99  939 
9  99938 
9  99938 

9  99  937 
9  99936 
9.99936 
9  99  935 
9  99  934 

9  99  934 
9  99  933 
9.99932 
9  99932 
9  99931 

9  99930 
9  99  929 
9.99929 
9.99928 
9.99927 

9.99926 
9.99926 
9-99925 
9-99924 
9.99923 

9.99923 
9.99922 
9  99921 
9.99920 
9.99920 

9.99919 
9.99918 
9.99917 
9  99917 
9.99916 

9  99915 
9.99914 

9  99913 
9  99913 
9.99912 

9  99911 
9  99910 
9.99909 
9.99909 
9.99908 

9,99907 
9.99906 
9  99905 
9  99904 
9  99  904 

9.99903 
9.99902 
9.99901 
9.99900 
9.99899 

9.99898 
9.99898 
9.99897 
9  99896 
9  99895 

9.99894 

L.  Sin. 

Prop.  Pts. 


238 

334 

sag 

.1 

23.8 

23 -4 

22 

.2 

47.6 

46. « 

45 

•3 

71.4 

70.2 

68 

•4 

95-2 

93-6 

91 

•5 

119.0 

117. 0 

114 

.6 

142.8 

140.4 

137 

•7 

166.6 

163.8 

160 

.8 

190.4 

187.2 

183 

•9 

214.2 

210.6 

206 

225 

220 

.1 

22.5 

22.0 

2 

45.0 

44.0 

•3 

67.5 

66.0 

•4 

90.0 

88.0 

•5 

112.5 

IIO.O 

.6 

135 -o 

132.0 

■7 

157-5 

I54-0 

.8 

180.0 

176.0 

9 

202.5 

198.0 

212 

208 

21.2 

20.8 

42.4 

41.6 

63.6 

62.4 

84.8 

83.2 

106.0 

104.0 

127.2 

124.8 

-7 

148.4 

145.6 

.8 

169.6 

166.4 

•9 

190.8 

187.2 

201 

197 

.1 

20.1 

19.7 

.2 

40.2 

39-4 

•3 

60.3 

59-1 

•4 

80.4 

78.8 

•5 

100.5 

98.5 

.6 

120.6 

118.2 

•7 

140.7 

137-9 

.8 

160.8 

157-6 

•9 

180.9 

J77-3 

i8g 
18.9 
37-8 
56.7 
75  6 
94-5 
"3  4 
132.3 
151.2 
170.1 


185 

18.5 
37-0 
55-5 
74 -o 
92.5 

III.O 

129.5 
148.0 
166. 5 


43a 

1  0.4  0.3  0.2  c 

2  0.8  0.6  0.4  o 

3  1.2  0.9  0.6  o 

4  1.6  1.2  o.'^  o 

5  2.0  1.5  1.0  o 

6  2.4  1.8  1.2  o 

7  2.8  2.1  1.4  o 

8  3.2  2.4  1.6  o 

9  3.6  2.7  1.8  o 


216 
21.6 
43-2 
64.8 
86.4 
108.0 
129.6 
151.2 
172.8 
194.4 
204 
20.4 
40.8 
61.2 
81.6 
102.0 
122.4 
142.8 
163.2 
183  6 
193 
19-3 
38.6 

57-9 

77.2 

96.5 

1158 

135  I 

154-4 

173-7 

181 

18. 1 

36.2 

54-3 
72.4 
905 
108.6 
126.7 
144.8 
162.9 


Prop.  Pts. 


30 


TABLE  II 


_9 
10 
II 

12 
13 

ii_ 

\i 

18 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
il 

36 
37 

3« 

40 

41 
42 

43 
44 

45 
46 

47 
48 

49 

50 

51 
52 
53 

54 

55 
56 
57 
58 
59 
60 


L.  Sin. 


8.84358 

8  84539 
8.84  718 
8  84897 
8  85  075 


8  85  252 
8.85429 
8  85  605 
8  85  780 
8  85955 


8  86  128 
8  86  301 
8  86  474 
8  86  645 
8  86816 


8  86  987 
8  87  156 

8  87325 
8  87494 
8  87661 


8  87  829 
8  87  995 
8  88  161 
8  88  326 
8  88  490 


8  88  654 
8  88817 
8.88980 
8  89  142 
8  89  304 


8  89  464 
8  89  625 
8  89  784 
8.89943 
8  90  102 


8  90  260 
8  90  417 
8  90574 
8  90  730 
8  90  885 


8  91  040 
8  91  195 
8  91  349 
8  91  502 
8  91  655 


8  91  807 
8  91.959 
8  92  no 
8  92  261 
8  92  411 


8  92  561 
8  92  710 
8  92  859 
8  93  007 
8  93  ^54 


8  93  301 
8  93448 
8  93  594 
8.93  740 
8  93  885 


8.94030 


L.  Cos. 


179 
179 
178 
177 

177 
176 
>75 
175 
173 
173 
173 
171 
171 
171 
169 
169 
169 
167 
168 
166 
166 
165 
164 
164 
163 
163 
162 
162 
160 
161 
159 
159 
159 
158 

157 
157 
156 
155 
'55 
155 
154 
153 
153 
152 
152 
151 
151 
150 
150 
149 
149 
148 
147 
147 

147 
146 
146 
H5 
145 


L.  Tang. 


8.84464 
8.84646 
8.84826 
8.85006 
8.85  185 


c.d. 


8  85  363 

8.85  540 

8.85717 
8.85893 

8.86  o6q 


8  86  243 
8.86417 
8.86591 
8.86  763 
8.86935 


8.87  106 
8.87277 

8.87447 
8.87616 

8  87  785 


8.87953 
8.88  120 
8.88287 
8.88453 
8.88618 


8.88783 
8.88948 
8.89  III 
8.89274 
8.89437 


8  89598 
8.89  760 
8 .  89  920 
8 .  90  080 
8 .  90  240 


8.90399 

8  90557 
8  90715 
8.90872 
8  91  029 


8.91  185 
8.91  340 
8.91  495 
8  91  650 
8.91  803 


91  957 
o  92  no 
8.92  262 
8.92  414 
8.92  565 


8.92  716 
8.92866 

8.93  016 
8.93  165 
8  93  S13 
8  93  462 
8  9^  609 
8  93  756 
8  93903 
8  94  049 


8  94  '95 


182 
180 
180 
179 
178 
177 
177 
176 
176 
174 

^74 
174 
172 
172 
171 
171 
170 
169 
169 
168 
167 
167 
1 66 
165 
165 

165 
163 
163 
163 
i6i 
162 
160 
160 
160 
159 
158 
158 
157 
157 
156 

155 
155 
155 
153 
154 
153 
152 
152 
151 
151 
150 
ISO 
149 

I  J*8 
149 

I  X47 
147 
147 

I  146 

j  146 


L.  Cotg. 


I  15  536 
I  15  354 
I  15  174 
1 .  14  994 
I  14  815 


1. 14  637 
I  14  460 
I  14  283 
I  14  107 
I  13  931 


13757 
13583 
13409 
13237 
13065 


1 .  12  894 
I  12  723 
I  12  553 
I  12  384 
1 .  12  215 


12047 
n  880 
II  713 
II  547 
11382 


I  II  217 
III  052 

1 .  10  889 
1 .  10  726 
I  10  563 


1 .  10  402 
1 .  10  240 
1 .  10  080 
1 .  09  920 
I . 09  760 


I  09  601 
1.09443 
I  09  285 
I .09  128 
1.08  971 


I  08815 
1 .08  660 
1.08505 
I  08350 
I .08  197 


I  08  043 
1.07  890 
1.07738 
1.07586 
I  07  435 
1 .07  284 
I  07  134 
I  06  984 
1.06  835 
1.06687 
To6~538" 
1.06  391 
I  06  244 
1 .06  097 
I  05951 
I  05  805 


L.  Cos. 


L.  Cotg.  ic.  d.l  L.  Tang. 

85^ 


99894 

99893 
99892 
99891 
99891 


99  890 
99889 
99888 
99887 
99886 


99885 
99884 
99883 
99882 
99881 


99880 
99879 
99879 
99878 
99877 


99876 
99875 
99874 
99873 
99872 


99871 
99  870 
99  869 
99868 
99867 


99866 
99865 
99  864 
99863 
99  862 


.  99  861 
9  99  860 
.  99859 
9  99  858 
9  99  857 


9  99  856 

9  99855 

99854 

99853 

99852 


99851 
99850 
99848 
99847 
99  846 


99845 
99844 
99843 
99842 

99841 


99  840 

99839 
99838 

99837 
99836 


99834 


L.  Sin. 


GO 

59 

58 
57 

55 
54 
53 
52 
51 
50 
49 
48 
47 

45 

43 
42 
41 
40 
39 
38 

36 

35 
34 
33 
32 
31 
30 
29 
28 

27 
26 


Prop.  Pts. 


181 

179 

I 

18.1 

17.9 

.2 

36.2 

35-8 

•3 

54-3 

53-7 

•4 

72.4 

71.6 

•5 

90-5 

89.5 

.6 

108.6 

107.4 

•7 

126.7 

125.3 

.8 

144.8 

143-2 

•9 

162.9 

161. 1 

175 

X73 

.1 

17-5 

17-3 

.2 

35.0 

34-6 

•3 

52-5 

51-9 

•4 

70.0 

69.2 

•5 

87.5 

86.5 

.6 

105.0 

103.8 

■7 

122.5 

121. 1 

.8 

140.0 

138.4 

•9 

157-5 

155.7 

168 

166 

I 

.16.8 

16.6 

3 

33-6 

33-2 

•3 

50-4 

49-8 

•4 

67.2 

66.4 

•5 

84.0 

83.0 

.6 

100.8 

99-6 

•7 

117. 6 

116.2 

.8 

134-4 

132.8 

•9 

151-2 

149.4 

162 

159 

16.2 

159 

32-4 

31-8 

48.6 

47-7 

64.8 

63.6 

81.0 

79-5 

6 

97-2 

95-4 

•7 

"3-4 

III. 3 

.8 

129.6 

127.2 

•9 

145.8 

I43-I 

155 

153 

I 

15  5 

153 

2 

31-0 

30.6 

3 

46.5 

45-9 

4 

62.0 

61.2 

5 

77-5 

76.5 

.6 

93-0 

91.8 

•7 

108.5 

107.1 

.8 

124.0 

122.4 

•9 

139-5 

'37-7 

149 

147 

14.9 

M-7 

29.8 

29.4 

44-7 
59-6 

44.1 
58.8 

.6 

74-5 
894 

73-5 
S8.a 

•7 
.8 

104  3 
119.2 

102.9 
1176 

•9 

134 -I 

132  3 

Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


31 


2 

3 
_4 

5' 
6 

7 
8 

9 

10 

II 

12 

13 

14 

15 
i6 

17 
i8 

20 

21 
22 
23 

26 

27 
28 

29_ 

30 

31 
32 

33 
34 

36 
37 
38 
39 
40 

41 

42 

43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
J9 
60 


L.  Sin. 


94030 
94  174 
94317 
94461 
94603 


94  746 
94887 
95029 

95  170 
95310 


95450 
95589 

95  728 
95867 

96  005 


96  143 
96  280 
96417 

96553 
96  689 


96825 
96  960 

97095 
97229 

97363 


97496 
97629 
97762 
97894 
98  026 


98157 
98288 
98419 

98549 
98679 


98808 

98937 
99  066 

99  194 
99322 


99450 
99  577 
99  704 
99830 
99956 


00  082 
00  207 
00332 
00  456 
00  581 


00  704 
00828 
00951 

01  074 
01  196 


01  318 
01  440 
01  561 
01  682 
01  803 


01  923 


L.  Cos. 


L.  Tang. 


94195 
94340 
94485 
94630 

94  773 


94917 
95  060 
95  202 
95  344 
954S6 


95627 
95  767 

95  908 

96  047 
96  187 


96325 
96  464 
96  602 
96  739 
96877 


97013 
97  150 
97285 
97421 
97556 


97691 
97825 

97  959 

98  092 
98  225 


98358 
98  490 
98  622 

98753 
98884 


99015 
99  145 
99275 
99405 
99  534 


99  662 

99  791 
99919 
00  046 
00  174 


00  301 
00  427 

00553 
00  679 
00  805 


00  930 

01  055 
01  179 
01  303 
01  427 


01  550 
01  673 
01  796 

01  918 

02  040 


02  162 


Cotg. 


c.d. 


c.d. 


L.  Cotg. 


1.05  805 
1 .  05  660 

I  05  515 

1.05  370 

05  227 


05  083 
04  940 
04  798 
04  656 
04514 


04373 
04233 
04  092 

03953 
03813 


03675 
03536 
03398 
03  261 
03  123 


02  987 
02  850 
02  715 
02579 
02444 


02  309 
02  175 
02  041 
01  908 
01  775 


01  642 
01  510 
01378 
01  247 
01  116 


1 .  00  985 
00  855 
I  00  725 
I  00595 
I  00  466 


I  00338 
1 .  00  209 
I  00  081 

o  99  954 
o  99  826 


0.99699 
o  99  573 
0.99447 
0.99321 
0.99  195 


o .  99  070 
o  98945 

0.98821 
0.98  697 

0.98573 


o .  95  450 

0.98327 

o .  98  204 
o .  98  082 
0.97  960 


0.97 


L.  Tang. 

84° 


I 

.Cos. 

60 

It 
11 

55 
54 
53 
52 
51 

9 
9 
9 
9 
9 

99834 
99833 
99832 
99831 
99830 

9 
9 
9 
9 
9 

99829 
99828 
99827 
99825 
99824 

9 
9 
9 
9 
9 

99823 
99822 
99821 
99  820 
99819 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 
40 

1 

9 
9 
9 
9 
9 

99817 
99  816 
99815 
99814 
99813 

9 
9 
9 
9 
9 

99812 
99  810 
99809 
99808 
99807 

9 
9 
9 
9 
9 

99806 
99804 
99803 
99802 
99801 

35 
34 
33 
32 
31 
30 

29 
28 

11 

9 
9 
9 
9 
9 

99800 
99798 

99  797 
99796 

99  795 

9 
9 
9 
9 
9 

99  793 
99792 
99791 
99790 
99788 

25 
24 
23 
22 
21 

9 
9 
9 
9 
9 

997S7 
99786 
99785 
99783 
99782 

20 

19 
18 

\l 

IS 
14 
13 
12 
II 

9 
9 
9 
9 
9 

99781 
99780 
99778 
99777 

99776 

9 
9 
9 
9 
9 

99  775 
99  773 
99772 
99771 
99769 

10 

7 
6 

9 
9 
9 
9 
9 

99768 
99767 
99765 
99  764 
99763 

5 
4 
3 
2 
I 
0 

9 

99761 

I 

.Sin. 

f 

Prop.  Pts. 


145 

X43 

141 

14.5 

14.3 

14. 

ap.o 

28.6 

28. 

43-5 

42.9 

42 

58.0 

57-2 

56 

72-5 

71-3 

70 

6 

87.0 

85.8 

84 

7 

101.5 

100. 1 

98. 

8 

116.0 

114.4 

112. 

9 

130.5 

128.7 

126. 

139 

,^3-9 
27 

41 

55 
69 

83 
97 


138 

13  ■ 
27 

41 

55 

69 

82 

96 
no 
.•24 


135 

133 

.1 

13.5 

13.3 

3 

27.0 

26.6 

3 

40.5 

39-9 

4 

540 

53-2 

5 

675 

66.5 

6 

81.0 

79.8 

7 

94-5 

93-1 

8 

108.0 

106.4 

9 

121.5 

119.7 

139 

128 

.1 

12.9 

12.8 

.2 

25.8 

25.6 

•3 

38.7 

38.4 

•4 

51.6 

51.2 

•5 

645 

64.0 

.6 

77-4 

76.8 

•7 

90-3 

89.6 

.8 

103.2 

t02.4 

•9 

116. 1 

115    2 

135 

133 

.t 

"•5 

12.3 

.2 

25 

0 

24.6 

3 

37 

5 

36.9 

4 

50 

0 

49.2 

•5 

62 

5 

61.5 

6 

75 

0 

73-8 

7 

87 

5 

86.1 

8 

100 

0 

98.4 

9 

112 

5 

110.7 

X3I 

xao 

.1 

12. 1 

12.0 

.2 

24.2 

«4.o 

•3 

36.3 

36.0 

.4 

48.4 

48.0 

•5 

60.5 

60.0 

.6 

72.6 

72.0 

7 

847 

84.0 

.8 

96.8 

96.0 

•9 

108.9 

108.0 

136 

13 
27 
40 

54 


131 
13-1 
26.2 
39-3 
52.4 
65.5 
78.6 
91.7 
104.8 
117.9 

136 

12.6 
25.2 
37-8 
50.4 
63.0 
75-6 
88.2 
100.8 
"34 
122 
12.2 
24.4 
36.6 
48.8 
61.0 

73-2 

85.4 

97.6 
109.8 

z 

0.1 

0.2 

0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 


Prop.  Pts. 


32 


TABLE  II 


6^ 


__9_ 
10 
II 

12 

13 
14 

15 

i6 

17 

i8 

19 
20 

21 

22 
23 

il 

26 
27 
28 
29 


31 

32 

33 
34 

36 

37 
38 
39 
40 

41 
42 

43 
44 

46 

47 
48 

49 

50 

51 

52 
53 

il. 

55 
56 

57 
58 
i9_ 
60 


L.  Sin. 


01  923 

02  043 
02  163 
02  283 
02  402 


02  520 
02  639 
02  757 
02  874 
02  992 


03  109 
03  226 
03342 
03458 
03574 


03  690 
03  805 

03  920 
04034 

04  149 


04  262 
04376 
04  490 
04  603 
04  715 


04828 

04  940 

05  052 
05  164 
05275 


05386 
05497 
05  607 

05  717 
05  827 


05937 
06  046 

06155 
06  264 
06372 


06  481 
06  589 
06  696 
06  804 
06  911 


07018 
07  124 
07231 

07337 
07442 


07548 
07653 
07758 
07863 
07968 


08  072 
08  176 
08280 
08383 
08486 


08589 


L.  Cos. 


d.  L.  Tang. 


02  162 
02  283 
02  404 
02  525 
02  645 


02  766 
02885 
03005 

03  124 
03  242 


03  361 
03479 
03597 
03  714 
03832 


03948 
04  065 
04  181 
04297 
04413 


04  528 
04  643 
04758 
04873 
04987 


05  lOI 
05214 
05  328 
05441 
05553 


05  666 
05  778 

05  890 

06  002 
06  113 


06  224 
06335 
06445 
06  556 
06666 


06775 
06885 

06  994 

07  103 

07  211 


07  320 
07428 
07536 
07643 
07751 


07858 
07964 
08  071 
08  177 
08  283 


08389 

08495 
08  600 
08  705 
08810 


08  914 


d«     L.  Cotg. 


c.  d. 


c.  d, 


L.  Cotg. 


0.97838 
0.97717 
0.97  596 
0.97475 
0.97355 


0.97234 
0.97  115 
0.96995 
0.96  876 
0.96  758 


0.96639 
0.96  521 
o .  96  403 
o .  96  286 

0.96  168 


0.96  052 

0.95935 

o  95  819 
0.95  703 
o  95587 


0.95472 

o  95  357 
0.95  242 
0.95  127 
o  95013 


0.94899 
0.94  786 
0.94  672 

o  94  559 
0.94447 


0.94334 
o  94  222 
0.94  no 

0.93998 
0.93887 


o  93  776 
0.93665 

o  93  555 
o- 93  444 

o  93  334 


0.93  225 

o  93  "5 
o .  93  006 
0.92  897 
0.92  789 


0.92  680 
0.92  572 
0.92  464 
0.92357 
0,92  249 


0.92  142 
o .  92  036 
0.91  929 
0.91  823 

0.91  717 


0.91  oil 

o  91  505 
0.91  400 
0.91  295 
0.91  190 


0.91  086 


L.  Tang, 

83° 


L.  Cos. 


99  761 
99760 

99  759 
99  757 
99756 


99  755 
99  753 
99  752 
99  751 
99  749 


99  748 
99  747 
99  745 
99  744 
99  742 


99741 
99  740 
99  738 
99  737 
99736 


99  734 
99  733 
99731 
99  730 
99  728 


99727 
99  726 

99  724 
99  723 
99721 


99  720 
99718 
99717 
99  716 
99714 


99  713 
99  711 
99710 
99  708 
99707 


99705 
99  704 
99702 

99  701 
99699 


99  698 
99  696 

99695 
99693 
99  692 


9.99690 
9.99689 
9.99687 
9 .  99  686 
9  99684 


99683 
99  681 
99  680 
99678 
99677 


9  99675 


L.  Sin. 


Prop.  Pts. 


lai 

xao 

I 

12.1 

12.0 

.3 

24.2 

24.0 

3 

36.3 

36.0 

4 

48.4 

48.0 

5 

60.5 

60.0 

6 

72.6 

72.0 

7 

84.7 

84.0 

8 

96.8 

96.0 

9 

108.9 

108.0 

118 

117 

.1 

II. 8 

II. 7 

.2 

236 

23 

4 

.3 

35-4 

35 

I 

•4 

47.2 

46 

8 

•5 

59.0 

58 

5 

.6 

70.8 

70 

2 

•7 

82.6 

81 

9 

.8 

94-4 

93 

6 

■9 

106.2 

los 

3 

"5 

114 

.1 

"•5 

II. 4 

.2 

33.0 

22.8 

•3 

34-5 

34-2 

'4 

46.0 

45-6 

.5 

57-5 

570 

.6 

69.0 

68.4 

•  7 

80.5 

79.8 

.8 

92.0 

91.2 

.9 

103.5 

102.6 

iia 

III  1 

.1 

II. 2 

II. I 

.2 

22 

4 

22.2 

•3 

33 

6 

33-3 

.4 

44 

8 

44-4 

•5 

56 

0 

55-5 

.6 

67 

2 

66.6 

•7 

78 

4 

77-7 

.8 

89 

6 

88.8 

•9 

100 

8 

99.9 

109 

108 

lOJ 

.1 

10.9 

10.8 

10 

.3 

31 

8 

21.6 

21 

•3 

32 

7 

32.4 

32 

•4 

43 

6 

43-2 

42 

•5 

54 

5 

54 -o 

53 

.6 

65 

4 

64.8 

64 

•7 

76 

3 

75.6 

74 

.8 

87 

2 

86.4 

85 

•9 

98 

I 

97.2 

96 

X06 

IC.5 

I 

10.6 

10.5 

.3 

21.2 

21. 0 

•3 

31-8 

31  5 

•4 

42.4 

42.0 

•5 
.6 

53  0 
63.6 

52  5 
63.0 

.7 
.8 

74.2 
84.8 

73-5 
84.0 

•9 

95-4 

94-5 

Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


33 


L.  Sin. 


9_ 
10 
II 

12 

13 

\l 

17 
i8 

19 

20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

30 

31 

32 
33 
34 

36 

37 
38 
39 
40 

41 

42 

43 
44 

46 

47 
48 

49 

50 

51 

52 
53 

II 
II 

59 
GO 


08589 
08  692 

08795 
08897 

08  999 

09  lOI 

09  202 
09304 
09405 
09  506 


09  606 
09  707 
09  807 
09907 
0006 


o  106 
o  205 

0304 

0402 
o  501 


0599 
0697 
0795 
0893 

o  990 


I  087 

1 184 
1 281 
1377 

I  474 


I  570 
I  666 
I  761 
1857 
1952 


2047 
2  142 
2  236 

2331 

2425 


2519 
2  612 
2  706 

2  799 
2  892 


2985 
3078 

3  171 
3263 

3  355 


3  447 
3  539 
3630 
3  722 
3813 


3904 

3  994 
4085 

4175 

4  266 


4356 


L.  Cos. 


103 

103 
102 
102 
1 03 
101 
1 03 

lOI 
lOI 

100 

lOI 

100 
100 

•99 
100 

99 
99 
98 

99 
98 

98 
98 
98 
97 
97 
97 
97 
96 

97 
96 

96 
95 
96 

95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
92 
92 
92 

91 
92 

91 
91 
90 

91 
90 

91 
90 


L.  Tang. 


c.  d. 


08  914 

09  019 
09  123 
09  227 
09330 


09434 
09537 
09  640 

09  742 
09845 


09947 
o  049 
o  150 
o  252 

0353 


0454 

o  656 

0756 
0856 


0956 

I  056 

1 155 
1 254 

I  353 


452 

649 
747 
845 


1943 
2  040 
2138 
2235 
2332 


2428 

2525 
2  621 

2  717 
2813 


2  909 
3004 
3099 

3  194 
3289 


3384 
3478 

3  573 
3667 

3  761 


3854 
3948 
4041 

4  134 
4227 


4320 
4412 
4504 
4  597 
4688 


4780 


L.  Cotgr. 


c.  d. 


L.  Cotg. 


0.91  086 
0.90  981 
0.90  877 

0.90  773 
0.90  670 


o .  90  566 
0.90463 
o .  90  360 
0.90  258 

0.90155 


0.90053 
0.89  951 
o .  89  850 
0.89  748 
o .  89  647 


0.89  546 

0.89445 
0.89344 

o .  89  244 
0.89  144 


o .  89  044 
o .  88  944 
0.88845 
0.88  746 
0.88647 


0.88548 
o .  88  449 
0.88351 
0.88253 
0.88  155 


0.88057 
0.87960 
c. 87  862 
0.87  765 
0.87668 


0.87572 

0.87475 
0.87379 
0.87283 
0.87  187 


0.87091 
0.86  996 
0.86  901 
0.86806 
0.86  711 


0.86616 
0.86  522 
0.86  427 
0.86333 
o .  86  239 


0.86  146 
0.86  052 
0.85959 
0.85  866 
0.85  773 


0.85  680 
0.85588 
0.85  496 
0.85403 
0.85  312 


0.85  220 


L.  Tang. 

82° 


L*  Cos. 


9.99667 
9 .  99  666 
9.99664 
9.99663 
9.99  661 


99675 
99674 
99672 
99670 
99  669 


99659 
99658 
99656 
99655 
99653 


99651 
99650 
99  648 
99647 
99645 


99643 
99642 
99  640 
99638 
99637 


99635 
99633 
99632 
99630 
99  629 


99627 
99625 
99624 
99  622 
99  620 


99  618 
99617 
99615 
99613 
99  612 


99  610 
99  608 
99607 
99605 
99603 


9.99  601 
9.99  600 
9-99  598 
99596 
99  595 


99  593 
99591 
99589 
99588 
99586 


99584 
99582 

99581 
99  579 
99  577 


9-99  575 


L.  Sin. 


60 

59 

58 

57 
_5i 
55 
54 
53 
52 
_5i 
50 

49 
48 

47 
_46 

45 
44 
43 
42 
41 
40 

39 
38 

_36 

35 
34 
33 
32 
_3i 
30 
29 
28 
27 
26 


Prop.  Pt8. 


105 

104 

I 

IO-5 

10.4 

.2 

21.0 

20.8 

•3 

31  5 

31.2 

•4 

42.0 

41.6 

.5 

52. s 

52.0 

.6 

63.0 

62.4 

•7 

73-5 

72.8 

.8 

84.0 

832 

•9 

94  5 

93  6 

102 

lOI 

I 

10.2 

10.  I 

2 

20.4 

20.2 

3 

•4 

i 

30.6 
40.8 

61 .2 

303 
40.4 

I 

l\:t 

£i 

■9 

9.. 8 

90.9 

9S 

) 

98 

I 

9  9 

9 

? 

19 

8 

19. 

•3 

29 

7 

29. 

•4 

39 

6 

39- 

49 

5 

49- 

.6 

59 

4 

58. 

7 

69 

3 

68. 

8 

79 

2 

78. 

9 

89 

I 

88. 

97 

9  7 
19 
29 
38 


48 
58 
67 
77 
87 

94 

9-4 
18.8 
28.2 
37  6 
47.0 

56.4 
65.8 
75-2 
84.6 


9e 

'  1 

9.6 

19.2 

28.8 

38.4 

48.0 

57-6 

67.2 

76.8 

86.4 

93 

9  3 

18 

6 

27 

9 

37 

2 

46 

55 

1 

65 

I 

74 

4 

83 

7 

103 

10.3 
20.6 

309 

41  .2 

6i'8 
72.1 
82.4 
92.7 
100 
10.0 
20.0 
30.0 
40.0 
50.0 
60.0 
70.0 
80.0 
90.0 


4 
2 

95 

9  5 
19  o 

285 
380 

47-5 


91 

90 

.1 

9  ^ 

9.0 

.2 

18 

2 

18.0 

•3 

27 

3 

27.0 

•4 

?6 

4 

36.0 

.5 

45 

5 

45  0 

.6 

54 

b 

.54  0 

•  7 

63 

7 

63.0 

.8 

r. 

8 

72.0 

9 

9 

81.0 

92 

9.2 
18.4 
27.6 

36.8 

46.0 

64.4 

73  6 
82.8 

a 

0.2 
0.4 
0.6 
0.8 
i.o 
1.2 

1-4 
1.6 
1.8 


Prop.  Pts. 


34 


TABLE  II 


8° 


L.  Sin, 


10 

II 

12 

13 

14 

15 

i6 

17 

i8 

i9_ 
20 

21 

22 
23 
24 

26 

27 
28 

30 

31 

32 
33 

34 

36 

37 
38 
39 
40 
41 
42 
43 
44 

46 

47 
48 

j49 

50 

51 
52 
53 
54 

59 
60 


E 


4356 
4445 
4535 
4624 

4  714 


4803 
4891 

4  980 
5069 

5  J57 


5245 
5  333 
5421 
5508 
5596 


5683 
5  770 
5857 
5  944 
6030 


6  116 
6  203 
6289 

6374 
6  460 


6545 
6631 
6  716 
6801 
6  886 


6970 
7055 
7  139 
7223 

7307 


7391 
7  474 
7558 
7641 
7724 


7807 
7890 
7  973 
8055 
8137 


6  220 

8  302 
8383 
8465 
8547 


8628 
8  709 
8790 
8871 
8952 


9033 
9  "3 
9  193 
9273 
9  353 


9  19433 


L.  Cos, 


d. 


89 


L.  Tan^. 


4780 
4872 
4963 
5054 
5  145 


5236 
5327 
5417 
5  "^08 
5598 


5  688 

5  777 
5867 

6  046 


6135 
6  224 
6312 
6  401 
6489 


6577 
6665 

6753 
6841 
6928 


7  016 
7103 
7  190 
7277 
7363 


7450 
7536 
7  622 
7708 
7  794 


7880 

7965 
8051 
8  136 
8221 


8306 
8391 
8475 
8560 
8644 


8728 
8812 
8896 
8979 
9063 


9  146 
9  229 
9312 

9  395 
9478 


9561 
9643 
9725 
9807 
9889 


9.19971 


L.  Cotg.  c.  d 


c.d. 


L.  Cotg. 


0.85  220 
o  85  128 
o  85037 
o  84  946 

0.84855 


o  84  764 
o . 84  673 

084583 

o .  84  492 
o .  84  402 


84312 
84223 
84133 

84  044 

83954 


83865 
83776 

83688 
83599 

83  5" 


0.83423 
o  83335 

0.83247 
0.83  159 

o . 83  072 


0.82984 
0.82897 
0.82810 

o  82  723 
o  82  637 


0.82  550 
o .  82  464 

0.82378 

0.82  292 
o  82  206 


0.82  120 
o  82  035 
o  81  949 
o  81  864 
o  81  779 


o  81  694 
0.81  609 
o  81  525 
o  81  440 

0.81  356 


0.81  272 

0.81  188 

0.81  104 
o  81  021 
o  80  937 


o  80  854 
o  80  771 
o  80688 
o .  80  605 
0.80  522 


o .  80  439 
0.80357 
0.80275 
0.80  193 
0.80  III 


.  80  029 


L.  T,aiig. 

8r 


L.  Cos. 


9  99  575 

9  99  574 
9  99  572 
9  99  570 
9  99  568 


99566 
99565 
99563 
99561 
99  559 


99  557 
99556 
99  554 
99552 
99550 


99548 
99  546 
99  545 
99  543 
99541 


99  539 
99  537 
99  535 
99  533 
99532 

99530 
99528 
99526 
99524 
99522 

99520 
99518 
99517 
99515 
99513 


99  5" 

99509 
99507 
99505 
99503 


99501 
99  499 
99  497 
99  495 
99  494 


99492 
99490 
99488 
99  486 
99484 


99482 
99  480 
99478 
99476 
99  474 


99472 
99470 
99  468 
99  466 
99464 


99462 


L.  Sin, 


GO 

59 
58 
57 
_5i 
55 
54 
53 
52 
_51 
50 
49 
48 
47 
_46^ 

45 
44 
43 
42 
41 
40 
39 
38 
37 
36 


Prop.  Pts. 


9i 

91 

1 

I 

9.2 

9.1I 

2 

18 

4 

2 

3 

27 

6 

27 

3 

4 

36 

8 

36 

4 

<; 

46 

0 

45 

5 

6 

55 

2 

S4 

6 

7 

64 

4 

63 

7 

8 

73 

6 

72 

8 

9 

82 

8 

81 

9 

90 

9.0 

18.0 

27.0 
36.0 

45.0 

54  o 
63.0 
72.0 
81.0 


89 


87 


17 
26 

34 
43 
52 
60 

69 
78 

85 


17 
25 
34 
42 
51 
59 
68 

76 

83 

8 
16 
24 
33 
41 
49 
58 
66 

74 


8.8 
17.6 
26.4 
35  2 
44  o 
52.8 
61.6 
70.4 
79  2 

86 

i.6 
17.2 
25.8 
34  4 
43  o 
51  6 
60.2 
68.8 
77  4 

84 

8.4 
16.8 
25.2 
33-6 
42.0 

50  4 
58.8 
67  2 
75.6 


16 
24 
32 
41 
49 
57 
65  6 

738 


81 

80 

I 

8.1 

8.0 

2 

16 

2 

16.0 

3 

24 

3 

24.0 

4 

32 

4 

32.0 

s 

40 

5 

40.0 

6 

48 

6 

48.0 

7 

S6 

7 

56.0 

8 

64 

8 

64.0 

9 

72 

9 

72.0 

3 
0.2 

0.4 
0.6 
0.8 

1 .0 

1.2 

1-4 
1.6 
1.8 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS         35 

9°                                        1 

0 

L.  Sin. 

d. 

L.  Tan^. 

c.d. 

L.  Cot^. 

L.  Cos. 

Prop.  Pts. 

9  19433 

80 

9  19  971 

82 

0  80  029 

9  99462 

(JO 

I 

9  19  513 

9.20053 

0  79  947 

9.99460 

59 

83       81       80 

2 

9  19592 

80 

9  20  134 

82 

0  79  866 

9  99  458 

58      .1 

8.2    8.1     8.0 

3 

9  19672 

79 
79 

9  20  216 

0  79  784 

9  99  456 

57       2 

16.4  16.2  16.0 

4 
5 

9  19  751 
9  19  830 

9  20  297 

81 
81 

0.79703 

9  99  454 

56      .3 
55      -4 

24.6  24.3  24.0 
32.8  32.4  32.0 

9  20  378 

0 . 79  622 

9  99  452 

6 

9  19909 

9  20  459 

0  79541 

9  99  450 

54      -5 

41 .0  40.5  40.0 

7 

9  19988 

79 

9  20  540 

0 .  79  460 

9  99  448 

53       ^ 

49.2  48.6  48.0 

8 

9  20  067 

79 
78 
78 

9  20  621 

80 
81 

0.79379 

9  99446 

52      •/ 

57.4  56.7  56.0 

9 
10 

9  20  145 

9  20  701 
9  20  782 

0  79  299 

9  99  444 

51      .^ 
50       5 

65  6  64.8  64.0 
73.8   72.9   72.0 

9  20  223 

0  79  218 

9  99  442 

II 

9  20  302 

78 

9  20  862 

80 

0  79  138 

9  99440 

49 

79 

78 

12 

9  20  380 

9  20  942 

0 . 79  058 

9  99438 

48 

.1       7.9 

7-8 

13 

9  20  458 

78 

9  21  022 

80 

0.78978 

9  99436 

47 

.2    15.8 

15.6 

14 

9  20535 

77 
78 

9.21  102 

80 

0.78898 

9  99  434 

46 
45 

.3    23.7 

4  31  6 

23  4 
31.2 

9  20  613 

9 

21  182 

0.78818 

9  99432 

i6 

9.20691 

78 

9 

21  261 

79 

0.78739 

9  99  429 

44 

5   39-5 

2^2 

17 

9  20  768 

77 

9 

21  341 

80 

0.78  659 

9  99427 

43 

6  47.4 

46.8 

i8 

9  ■  20  845 

77 

9 

21  420 

79 

0.78  580 

9  99  425 

42 

l  I^^ 

54  6 

19 
20 

9  20  922 

77 
77 

9 

21499 

79 
79 

0.78  501 

9  99423 

41 
40 

.8  63.2 
•9l  71  I 

62.4 
70.2 

9.20999 

9 

21578 

0.78422 

9  99  421 

21 

9.21  076 

77 

9 

21  657 

79 

0  78343 

9.99419 

39 

77 

76 

22 

9  21  153 

77 
76 

9 

21  736 

79 

0.78264 

9.99417 

38 

.1     7-7 

7.6 

23 

9.21  229 

9 

21  814 

78 

0.78  186 

9  99415 

37 

.2   15  4 

^5-^ 

24 

9.21  306 

77 
76 

9 

21893 

79 
78 

0.78  107 

9  99413 

36 
35 

■3  23.1 

.4  30.8 

22.8 
30.4 

2S 

9.21  382 

9 

21  971 

0 .  78  029 

9  99  411 

26 

9.21  458 

76 

9 

22049 

78 

0.77951 

9.99409 

34 

l  ^l^ 

38. 0 
45-6 

27 

9  21  534 

76 

9 

22  127 

78 

0.77873 

9.99407 

33 

.6  46.2 

28 

9.21  610 

76 

9 

22  205 

78 

0.77795 

9.99404 

32 

I  in 

9  693 

^:8 

68.4 

29 

30 

9.21  685 

75 
76 

9 

22  283 

78 
78 

0.77717 

9  99402 

31 
80 

9.21  761 

9 

22  361 

0.77639 

9.99400 

31 

9.21  836 

75 

9 

22438 

77 

0.77562 

9  99398 

29 

75 

74 

32 

9.21  912 

76 

9 

22  516 

78 

0.77484 

9.99396 

28 

.1    75 

7-4 

33 

9.21  987 

75 

9 

22593 

77 

0.77407 

9  99  394 

27 

.2  15.0 

14  8 

34 
35 

9  22  062 

75 
75 

9 

22  670 

77 
77 

0.77330 

9  99392 

26 
25 

•3  22.5 
•4  30  0 

22.2 
29.6 

9.22  137 

9 

22747 

0.77253 

9  99  390 

3^ 

9.22  211 

74 

9 

22  824 

77 

0.77176 

9  99  388 

24 

•5  37  5 

.6  45.0 

37  0 

37 

9  22  286 

75 

9 

22  901 

77 

0.77099 

9  99385 

23 

Tii 

3« 

9.22  361 

75 

9 

22977 

7b 

0.77023 

9  99383 

22 

.8  60.0 
■  9  67  5 

39 
40- 

9  22435 

74 
74 

9 

23054 

77 
76 

0.76946 

9  99381 

21 
20 

Iti 

9  22509 

9 

23  130 

0.76  870 

9  99  379 

41 

9.22583 

74 

9 

23  206 

76 

0.76  794 

9  99  377 

19 

73 

72 

42 

9  22657 

74 

9 

23283 

77 

0.76  717 

9  99  375 

18 

•  I     7-3 

7.2 

43 

9.22  731 

74 

9 

23359 

7b 

0.76  641 

9  99  372 

17 

.2   14.6 

144 

44 

9.22805 

74 
73 

9 

23435 

7b 
75 

0.76565 

9  99370 

16 
15 

.3  21.9 

4  29.2 

5  36.5 

6  43  8 

7  51  I 

.8  58  4 

•9  65.7 

21.6 
28.8 
36.0 
43-2 
50  4 
57  6 
64.8 

45 

9  22878 

9 

23510 

0.76490 

9.99368 

46 

9.22952 

74 

9 

23  586 

7b 

0.76414 

9.99366 

14 

'  47 

9  23025 

73 

9 

23661 

75 

0.76339 

9  99364 

13 

,4« 

9  23  098 

73 

9 

23737 

7b 

0 . 76  263 

9  99362 

12 

49 
60 

9  23  171 

73 
73 

9 

23812 

15 
75 

0  76  188 

9  99  359 

II 
10 

9 .  23  244 

9 

23887 

0.76  113 

999  357 

51 

9  23317 

73 

9 

23962 

75 

0.76038 

9  99  355 

?>                    T 

71 

3         3 

52 

9  23390 

73 

9 

24037 

75 

0.75963 
0.75888 

9  99  353 

8        I 

7.1     c 

.6     0.4 
.9     0.6 
.2     0.8 
5     '0 

53 

9.23462 

72 

9 

24  112 

75 

9  99351 

7      1 

14.2     c 

55 

9  23535 

73 
72 

9 

24  186 

74 
75 

0.75  814 

9-99  348 

6        3 

5       1 
4        6 

21.3     c 

28.4  I 

35-5     I 
42.6     I 

49  7     2 

56.8  2 

63.9  2 

9.23607 

9 

24  261 

0.75739 

9 -99  346 

5t> 

9  23679 

72 

9 

24335 

74 

0.75665 

9  99  344 

.8     1 .2 

H 

9  23  752 

73 

9 

24410 

75 

0.75590 

9  99342 

I      -7 

8 

.1     1.4 
.4     1.6 
7     I  8 

5^ 

9.23823 

71 

9 

24484 

74 

0  75516 

9  99340 

59 

60 

9  23895 

72 
72 

9  24  55« 

74 
74 

0.75442 

9  99  337 

'        0 

9.23967 

9.24632 

0.75368 

9  99  335 

0      ^ 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tan^. 

L.  Sin. 

t 

Prop.  Pts.        1 

80°                                        1 

36 


TABLE  II 


10^ 


_9_ 
10 
II 

12 

13 

14 

15 

16 

17 
18 

19 

20 

21 

22 
23 
^_ 

'25 

26 

27 

28 

29 
30 

31 

32 

33 

36 
37 
38 
39 
40 
41 
42 
43 
44 

46 

47 
48 

49_ 

50 

51 

52 
53 
51 
55 
56 
57 
58 
19. 

(io 


]j.  Siii< 


23967 
24039 
24  no 
24  181 
24253 


24324 
24395 
24  466 

24536 
24  607 


24677 
24  748 
24818 
24888 
24958 


25  028 
25  098 
25  168 
25237 
25  307 


d*  L.  Tan^.  c.  d 


25376 
25445 
25514 
25583 
25652 


25  721 
25  790 
25858 
25927 
25995 


26  063 
26  131 
26  r9'9 
26  267 
26335 


26403 
26  470 
26538 
26  605 
26  672 


26739 
26806 
26873 

26  940 

27  007 


27073 
27  140 
27  206 
27273 
27  339 


27405 
27471 

27537 
27  602 
27668 


27  734 

27  799 
27864 
27930 
27995 

28  060 


L.  Cos. 


72 

71 
71 
72 

71 

71 
71 
70 

71 
70 

71 
70 
70 
70 
70 
70 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 

68 
69 
68 
68 
68 
68 
68 
68 
68 

67 
68 
67 
67 
67 
67 
67 
67 
67 
66 

67 
66 
67 
66 
66 
66 
66 
65 
.  66 
66 

6S 
65 
66 
65 
65 


24  632 
24  706 
,24779 
24853 
24  926 


.25  000 

25073 
.25  146 
25  219 
25  292 


25365 
25437 
25510 
25582 

25655 


•25  727 

25  799 
.25  871 

•25943 
.26  015 


26086 
26  158 
26  229 
26  301 
26372 


26443 
26514 

26585 
26  655 
26  726 


26797 
26867 
26937 
27  008 
27078 


27  148 
27  218 
27288 
27357 
27427 


9 
9 
9 
9_ 
9 
9 
9 
9 
9. 
9 
9 
9 
9 
9_ 
9 
9 
9 
9 
9_ 
9 
9 
9 
9 
_9 
9 
9 
9 
9 
_9 
9 
9 
9 
9 
J9 
9 
9 
9 
9 

_9 

L.  Cot^. 


2,7496 
27  566 

27635 

27  704 

27773 


.27  842 
.27911 

27  980 

28  049 
28  117 


,28  186 
.28254 
28323 
28391 
,28459 


28  527 
28  595 
,28662 
28  730 
,28  798 
,28865 


74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 

73 
72 

73 
72 

72 
72 
72 
72 
71 
72 

71 
72 

71 
71 

71 
71 
70 

71 
71 
70 
70 

71 
70 
70 
70 
70 
69 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 

68 
69 
68 
69 
68 
68 
68 
68 
67 
68 
68 
67 

c.  d. 


L.  Cotg. 


0.75368 
0.75294 
0.75  221 

0.75  147 
0.75074 


o .  75  000 
0.74927 
0.74854 
0.74781 
o . 74  708 


0.74635 
0.74563 
0.74490 
0.74418 
0.74345 


0.74273 
0.74  201 
0.74  129 
0.74057 
0.73985 


0.73914 
0.73842 
0.73  771 
0.73699 
0.73  628 


0.73557 
0.73486 
0.73415 
0.73345 
0.73274 


0.73203 

0.73  133 
0.73063 
o .  72  992 
0.72  922 


L.  Cos. 


0.72  852 
0.72  782 
0.72  712 
0.72643 
0.72573 


o .  72  504 

0.72434 
0.72  365 

o .  72  296 
0.72  227 


0.72  155 
o . 72  089 
o . 72  020 
0.71  951 

0.71  883 


0.71  814 
0.71  746 

0.71  677 

0.71  609 

0.71  541 


0.71  473 
0.71  405 
0.71  338 
0.71  270 
0.71  202 

0.71  135 

L.  Tang. 

79^ 


9  99335 
9  99333 
9  99331 
9.99328 

9  99326 


9  99 
9  99 
9  99 
9  99 
9  99 


324 
322 

319 
317 
315 


9  99 
9.99 

9  99 
9  99 
9  99 


3^3 
310 
308 
306 
304 


9.99301 
9.99299 
9.99297 
9.99294 
9,99292 


9.99290 
9 .  99  288 
9.99285 
9.99283 
9.99  281 


9.99278 
9.99276 
9.99274 
9.99271 
9.99269 


9.99267 
9.99264 
9 .  99  262 
9 .  99  260 

9-99  257 


9  99255 
9.99252 
9.99250 
9.99248 
9  99245 


99243 
99241 
99238 
99236 
99233 


9.99231 
9.99229 
9.99  226 
9  99224 
9.99  221 


9.99219 
9.99217 
9,99214 
9.99  212 
9  99209 


9.99207 
9.99204 
9  99  202 
9 .  99  200 
9  99  197 
9  99  195 
L.  Sin. 


60 

59 
58 

56 


45 
44 
43 
42 

_1L 
40 

39 
38 

_3i 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 


25 
24 

23 
22 
21 

20^ 

^9 
18 

17 
16 


Prop.  Pis. 


74 

73 

I 

^i 

7. 

2 

14.8 

14 

•3 

22.2 

21. 

4 

29.6 

29 

.5 

37.0 

36. 

.6 

44.4 

43 

■7 

51.8 

51 

.8 

59.2 

58 

•9 

66.6 

165 

7a 

I 

7.2 

2 

14.4 

3 

21,6 

4 

28.8 

36. c 

.6 

43  2 

.7 

.50.4 

.8 

57-6 

■9 

64.8 

70 

7.0 

14,0 

21  .0 
28.0 

35  o 
42.0 
49  o 
56,0 
63.0 
68 

6.8 
13-6 
20.4 
27.2 
34  o 
40.8 
47.6 

54-4 
61.2 


3 

03 
0.6 

0.9 
I  .2 


66 

I 

6.6 

2 

13.2 

3 
4 

19.8 
26.4 

.7 
.8 

330 
39  6 
46.2 
52.8 

•9 

59  4 

4 
7 
71 
71 
14 
21 
28 

35 
42 

49  7 
56.8 

63  9 

69 

69 
13  '^ 

20, 

27 
34 
41 
48 

55 
62 

67 
6. 

13 

20. 
26. 

33 

40.2 
46.9 
53  6 
60.3 

65 

6.5 
13.0 

26.0 

32  5 
39  o 

45  5 
52.0 

58.5 
a 

0.2 
0.4 
06 
08 
1 .0 

12 

14 
16 

1.8 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS         37 


11 


7 
8 

_9_ 

10 

12 

13 

\l 

17 

i8 

19 
20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

30 

31 
32 

33 

34 

36 
37 
38 
39 
40 
41 
42 
43 
44 

46 

47 
48 

49 

60 

51 

52 
53 
ii 
55 
56 

U 

60 


L.  Sin. 


9  28  060 
9  28  125 
9.28  190 
9.28254 
9,28319 


9  •  28  384 
9  2S  448 
9.21  512 
9.28577 
9  28  641 


9.28  705 
9.28  769 
9  28  833 
9 .  28  896 
9  28  960 


9  29  024 
9.29  087 
9.29  150 
9.29  214 
9.29277 


9  29340 
9.29403 
9 .  29  466 
9.29529 
9.29591 


9.29654 
9.29  716 

9  29  779 
9.29841 
9.29903 


9 .  29  966 
9 .  30  028 
9.30090 

930  151 
9.30213 


9  30275 
9  30336 
9  30398 
9  30459 
9  30521 


9.30  582 
9  30643 
9,30  704 

9  30  765 
9 .  30  826 


9.30887 

9  30947 
9  31  008 
9.31  068 
9  31  ^29 
9.31  189 
9  31  250 
9-31  310 
9  31  370 
9  31  430 
9.31  490 

9  31  549 
9  31  609 
9,31  669 
9  31  728 


9.31  788 


L.  Cos. 


65 

65 
64 
65 
65 
64 
64 
65 
64 
64 
64 
64 
63 
64 
64 

63 
63 
64 
63 
63 
63 
63 
63 
62 
63 
62 

63 
62 
62 
63 
62 
62 
61 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 

6i 
60 
60 
60 
60 

59 
60 
60 

59 
60 


L.  Tang,  c.  d 


28865 

28933 
29  000 
29  067 
29  134 


29  201 
29  268 

29335 
29  402 
29  468 


29535 
29  601 
29668 

29734 
29  800 


29866 

29  932 
29998 

30  064 
30  130 


30195 
30  261 
30326 
30391 
30457 


30  522 

30587 
30652 
30717 
30  782 


30  846 

30  91 1 

30975 

31  040 
31  104 


31  168 
31  233 
31  297 
31  361 
31425 


•  31  489 

31  616 
31  679 

31  743 


31  806 
31  870 

31933 
31996 
32059 


32  122 
32185 
32248 

323" 

32373 


32436 
32498 

32561 
32623 
32685 


32747 


68 
67 
67 
67 
67 

67 
67 
67 
66 
67 
66 
67 
66 
66 
66 
66 
66 
66 
66 
65 
66 
65 
65 
66 
65 
65 
65 
65 
65 
64 

65 
64 
65 
64 
64 

65 
64 
64 
64 
64 

63 
64 
63 
64 
63 
64 
63 
63 
63 
63 
63 
63 
63 
62 

63 
62 
61 
62 
62 
62 


L.  Cotg. 


0.71  135 
0.71  067 
0,71  000 

o  70933 
o . 70  866 


0.70799 
o .  70  732 
o .  70  665 
0.70  598 
0.70532 


o .  70  465 

0.70399. 
0.70332 

o .  70  266 
o .  70  200 


0,70  134 

o .  70  068 
o .  70  002 

0.69936 

o  69  870 


o .  69  805 

0.69739 

0.69  674 
0.69  609 

0.69543 


0.69  478 

0.69413 
0.69348 

o .  69  283 
o  69  218 


0.69  154 
o .  69  089 
o .  69  025 
0.68  960 

0.68896 


L.  Cos. 


L.  Cotg.  Ic.  d. 


0.68832 
0.68  767 
0.68  703 
o  68  639 
0.68575 


0.68  511 
0.68448 
0.68384 
o  68321 
0.68  257 


0.68  194 
0.68  130 
0.68067 
o .  68  004 
o  67  941 


0.67878 
0.67815 
o  67  752 
o  67  689 
0.67  627 


0.67  564 
0.67  502 
o  67439 
0.67377 
o  67315 


67253 


L.  Tang. 

78^ 


9  99 195 
9.99  192 
9.99 190 
9.99 187 
9  99 185 


9.99 182 
9.99 180 
9.99  177 

9  99  175 
9.99  172 


9.99  170 
9.99  167 
9.99  165 
9  99 162 
9  99 160 


9  99 
9  99 
9  99 
9  99 
9  99 


157 
155 
152 
150 
147 


9  99 
9  99 
9  99 
9  99 
9  99 


145 
142 
140 
137 
135 


9  99 
9  99 
9  99 
9  99 
9  99 


132 
130 
127 
124 
122 


9  99 
9  99 
9  99 
9  99 
9  99 


9  99 
9  99 

9-99 
9  99 
9  99 


106 
104 

lOI 

099 
096 


9  99 
9  99 
9  99 
9  99 
9  99 


093 
091 
088 
086 
083 


9  99 
9  99 
9  99 
9  99 
9  99 


080 
078 

075 
072 
076 


9  99 
9  99 
9  99 
9  99 
9  99 


067 
064 
062 

059 
056 


9  99054 
9,99051 
9.99048 
9.99046 
9  99043 


9  99  040 


L.  Sin. 


60 

59 
58 

55 
54 
53 
52 

_5L 
50 

49 
48 
47 
46 


25 
24 
23 
22 
21 

20 

19 
18 

17 
16 


Prop,  rts. 


68 

6.8 
13.6 
20  4 
27  2 
34  o 
40.8 
47  6 

54-4 
61  2 

66 

6.6 
13  2 
10  8 
26 
33 
39 
46 
52 
59-4 

64 

6.4 
12.8 
19.2 
25.6 
32.0 
38.4 
44-8 
51  2 
576 

63 

62 


12 


24 
31 
37 
43 
49 
55 

60 

6.0 
12.0 
18.0 
24.0 
30,0 
36.0 
42  o 
48.0 
54  oi 


67 

07 
13  4 
20  I 
26  8 

33  5 
40.2 

46  9 

60.3 

65 

65 
13 
19 
26 

32 
39 
45 
52 
58 
63 

12.6 
189 
25  2 

3^  5 
37  8 
44  I 
50  4 
567 

61 

6 
12 


18 
24 

36 

42 

48 

54 

59 

5  9 
II 

17 

23 


3 

.1 

03 

.2 

0.6 

•3 

0.9 

4 

1.2 

I 

'A 

7 

2. 1 

8 

2.4 

9 

2.7 

a 

0.2 
0.4 
0.6 
0.8 
1.0 
1.2 

14 
16 
1.8 


Prop.  Pts. 


38 


TABLE  II 


12°                                         1 

/ 

L.  Sin. 

d. 

L.Taug.  c.d.| 

L.  Cotg. 

L.  Cos. 

60 

Prop.  Pts. 

0 

9.31  788 

9  32  747 

63 

0.67253 

9.99040 

9  31  847 

9.32  810 

0.67  190 

9.99038 

59 

63         6a 

2 

9.31  907 

9-32872 

0.67  128 

9  99035 

58 

I      63      6.2 

3 

9.31  966 

59 

9  32933 

62 

0.67067 

9.99032 

57 

.2     12   6     12   4 

_± 

9.32025 

59 

9-32995 

62 
62 

0.67005 

9.99030 

56 

55 

.3     18   9     18.6 
.4    25    2    24.8 

9.32084 

9-33  057 

0.66943 

9  99027 

6 

9  32  143 

59 

9  33  "9 

fi-W 

0.66881 

9  99  024 

54 

.5  31  5  31  0 

7 

9.32  202 

59 

9  33  180 

62 

61 

0.66820 

9.99022 

S3 

-6  37  8  37  2 

8 

9.32  261 

59 
58 
59 

9  33242 

0.66  758 

9.99019 

52 

-7  44-1   43.4 

9 
10 

9  32319 

9  33303 

62 

0.66697 

9.99016 

51 

.8   50.4  49  6 
•9  567  55-8 

9  32378 

9  33  365 

0.66635 

9.99013 

50 

II 

9  32437 

59 
58 
58 

9  33426 

61 

0.66  574 

9.99  on 

49 

61        60 

12 

9  32495 

9  33487 

61 

0.66  513 

9  99  008 

48 

.1     6.1     6.0 

13 

9  32553 

9  33548 

61 

0.66452 

9  99  005 

47 

.2    12.2   12  0 

14 
15 

9  32  612 

59 
58 

58 
58 
58 
58 
58 
58 

9.33609 

61 
61 
61 
61 
60 
61 

0.66  391 

9  99002 

46 

.3    18.3   18.0 
.4  24.4  24.0 

9  32  670 

9-33670 

0.66330 

9  99000 

45 

16 

9  32  728 

9  33  731 

0.66269 

9  98  997 

44 

I  ^2i  ^2  ° 

17 

932786 

9-33  792 

0.66208 

9  98  994 

43 

.6  36.6  36.0 

18 

9.32844 

9  33853 

0.66  147 

9  98991 

42 

.7  42.7  42.0 
.8  48.8  48.0 
•9  54-9  540 

19 

20 

9.32902 

9  33913 

0.66087 

998989 

41 
40 

9.32960 

9  33  974 

0 .  66  026 

9.98986 

21 

9.33018 

9  34034 

At 

0.65  966 

9-98983 

39 

59 

22 

9  33075 

57 

S8 

9  34095 

0.65905 

9.98980 

38 

•'     ^1 

2S 

9  33  133 

9-34155 

0.65  845 

9.98978 

37 

.2   II. 8 

24 
25 

9  33  190 

57 
58 

9  34215 

61 
60 
60 
60 
60 
60 

0.65  785 

9  98975 

36 

35 

.3   17  7 
•4  23.6 
•5  29-5 
•6  35  4 

9  33248 

9.34276 

0.65  724 

9  98972 

26 

9  33305 

57 

9  34336 

0.65  664 

9  98969 

34 

27 

9  33362 

57 
58 

9  34396 

0.65  604 

9.98967 

33 

28 

9-33  420 

9  34456 

0.65  544 

9  98964 

32 

•  7  41  3 
.8  47  2 

9  53  I 

58         57 

1  5.8      5.7 

2  II. 6   II. 4 

29 

30 

9-33  477 

57 
57 

9  34516 

0.65  484 

9.98  961 

3^ 
30 

9-33  534 

9  34  576 

0.65  424 

9.98958 

V 

9  33591 

57 

9  34635 

59 
60 
60 

0.65  365 

9  98955 

29 

32 

9-33  647 

56 

9  34695 

0.65  305 

9  98953 

28 

33 

9  33  704 

57 

9  34  755 

0.65245 

9.98950 

27 

34 

9-33761 

57 
57 
56 

9  34814 

59 
60 

0.65  186 

9.98947 

26 

,3    17.4    171 
.4  23.2   22.8 
.5   29.0   28,5 
.6  34.8   34  2 
7  40  6  39  9 
.8  46.4  45.6 
.9  52  2   51.3 

56         55 

•I     56     55 

3S 

9  33818 

9  34874 

0.65  126 

9  98944 

25 

3^ 

9  33874 

9  34  933 

59 

0.65  067 

9.98941 

24 

37 

9  33931 

57 
56 
56 
57 

9  34992 

59 

0.65  008 

9.98938 

23 

3« 
39 
40 

9  33987 
9 -34  043 

9  35051 
9  35  "I 

59 
60 

59 

0.64949 
0.64889 

9-98936 
9  98933 

22 
21 
20 

9  34  100 

9  35  170 

0.64830 

9.98930 

41 

9  34156 

56 

9  35229 

59 

0.64  771 

9.98927 

19 

42 

9.34212 

56 

9  35288 

59 

0.64  712 

9  98924 

18 

43 

9 .  34  268 

56 
56 
56 
56 

9  35  347 

59 
58 
59 

0.64653 

9.98  921 

17 

3   16. 8i  16  5 

.4  22  4  22.0 
,5  28  0  27.5 
.6  33.6  33.0 
-7  39-2  38.5 
8  44.8  44.0 

44 
45 

9  34324 

9  35  405 

0.64595 

9  98919 

lb 
15 

9-34380 

9  35  464 

0  64  536 

9  98  916 

46 

9-34436 

9  35  523 

59 
58 

0.64477 

9  98913 

14 

47 

9-34  491 

55 

9  35  581 

0.64419 

9  98  910 

»3 

48 

9-34  547 

56 

9-35  640 

59 
58 
59 
58 
58 
58 
58 
S8 
58 
58 
58 
58 
57 

0 .  64  360 

9  98907 

12 

49 
50 

9.34602 

55 
56 

935698 

0 .  64  302 

9.98904 

II 
10 

.9  50  4  49  5 

3           3 

9-34658 

9  35  757 

0.64243 

9.98901 

51 

9-34713 

55 

9  35815 

0.64  185 

9.98898 

9 

I       0.3      0.2 
.2      0.6      0.4 

3      0.9      0.6 
.4       1.2      08 

•5     15     10 
.6     1.8     1.2 

52 

9-34769 

56 

9  35873 

0.64  127 

9.98896 

8 

S3 

9.34824 

55 

9  35931 

0 .  64  069 

9.98893 

7 

54 

9  34879 

55 
55 

9  35989 

0.64  on 

9.98890 

6 

5 

9  34  934 

9.36047 

0.63953 

9  98887 

56 

9.34989 

55 

9  36  105 

0  63  895 

9  98  884 

4 

57 

9-35044 

55 

9.36  163 

0.63837 

9.98881 

3 

7     2.1      14 

S« 

9-35099 

55 

9.36221 

0.63  779 

9.98878 

2 

.8     24     16 

59 

or 

9  35  154 

55 
55 

9.36279 

0.63  721 

9.98875 

I 

.9     2.7     1.8 

9  35209 

9  36336 

0 .  63  664 

9.98872 

0 

L.  Cos. 

d.  |l.  Cotg. 

c.d. 

L.  Taug. 

L.  Sin. 

Prop.  Pts. 

77°                   1 

LOGARITHMS  OF  THE  TKIGONOMETRIC  FUNCTIONS 


39 


13°                                         1 

0 

L.  Sin. 

54 
55 
55 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

60 

Prop.  Pts.        1 

9  35209 

9  36  336 

58 
58 
57 

0.63  664 

9.98872 

1 

I 

9 

35263 

9  36  394 

0.63  606 

998869 

S9 

58 

57 

2 

9 

35318 

9  36452 

0  63  548 

9  98867 

58 

•  I     5.^ 

?     5  7 

3     II    4 

\     17    I 
I     22    8 

3 

9 

35  373 

9  36  509 

0  63  491 

9.98864 

57 

.2  II. ( 

4 

9 

35427 

54 

9.36566 
9.36624 

58 

0  63434 

9  98861 

56 

S5 

•3   17- 
•4  23. i 

9 

35481 

0  63  376 

9  98858 

6 

9  35  536 

9.36681 

0.63319 

9  98855 

S4 

•5   29.: 

D    28   5 

7 

9  35  590 

9  36738 

0  63  262 

9  98  852 

SS 

.6  34. { 

^   34  2 

8 

9  35644 

9  36  795 

0  63  205 

9  98  849 

S2 

.7  40  6   39.9       1 

9 
10 

9 
"9 

35  698 

54 

9.36852 

57 

0  63  148 

9 .  98  846 

51 
50 

.8  46  . 
•9  52.^ 

\  45-6 
2  51  3 

35752 

9  36909 

0.63091 

9  98843 

II 

9  35  806 

9  36966 

0.63034 

9  98  840 

49 

56 

55 

12 

9.35860 

9  37023 

57 

0.62977 

9  98837 

48 

.1     5  < 

^     5  5 

13 

9 

35914 

9  37  080 

0.62  920 

9.98834 

47 

.2   II.: 

2   II. 0 

14 
IS 

9 

35968 

54 

9  37  137 

57 
56 

0  62863 

9  98  831 

46 
45 

.3   16.8   16.5 
.4  22.4  22.0 

9 

36022 

9  37  193 

0.62  807 

9.98828 

i6 

9 

36075 

53 

9  37250 

57 
56 

0.62  750 

9  98825 

44 

•5  28.0  27  5 

17 

9 

36  129 

9  37306 

0 .  62  694 

9.98822 

43 

•6  33( 

^  330 

i8 

9 

36  182 

53 

9  37363 

56 
57 
56 
56 
56 
56 
56 

0.62  637 

9.98819 

42 

■9  50-^ 

\  38.5 

19 
20 

9 

36236 

53 

9  37419 

0.62  581 

9.98816 

41 
40 

5  44.0 
^  49-5 

9 

36289 

9  37  476 

0.62  524 

9.98813 

21 

9 

36342 

9  37532 

0.62468 

9.98810 

39 

54 

22 

9 

36395 

937588 

0.62  412 

9.98807 

38 

.1 

5  4 

23 

9 

36449 

54 

9  37644 

0.62  356 

9.98804 

37 

.2   10.8            1 

24 
2S 

9 

36502 

53 

9.37700 

0.62  300 

9.98801 

36 

35 

•3   I 
•4  2 

62 
16 

9  36555 

9  37756 

0  62  244 

9.98798 

26 

9  36  608 

53 

9  37812 

56 

0.62  188 

9  98  795 

34 

•5  27.0 

27 

9  ^6  660 

52 

9  37  868 

56 

0.62  132 

9  98  792 

33 

t  ^li 

28 

9  36713 

53 

9  37924 

56 
56 
55 

0.62  076 

9.98789 

32 

■I  V 

29 

9  36766 

53 
53 

9  37980 

0 .  62  020 

9  98  786 

31 
30 

.8  4 
•9  4 

3  2 
8.6 

9 

36819 

9  38035 

0.61  965 

9.98783 

31 

9 

36871 

52 

9.38091 

56 

0.61  909 

9 . 98  780 

29 

53 

5a 

32 

9 

36924 

53 

938  147 

50 

0.61  853 

9.98  777 

28 

•I     5' 

;    52 

33 

9 

36976 

52 

9  38  202 

55 

0.61  798 

9.98  774 

27 

.2  10. ( 

)  10.4 

34 

9 

37028 

52 
53 

9  38257 

55 
56 

0.61  743 

9.98771 

26 

•3   15  S 

)  156 
20.8 
26.0 

35 

9 

37081 

9  38313 

0  61  687 

9.98768 

25 

.4  21.2 
•  5  26. c 
.6  31-^ 

3^ 

9 

37  133 
37185 

52 

9  38  368 

55 

0  61  632 

9.98765 

24 

37 

9 

52 

9  38  423 

55 

0.61  577 

9.98  762 

23 

;  31.2 

36  4 

\  416 

'  46.8 

3« 

9 

37  237 

52 

9  38  479 

56 

0.61  521 

9  98  759 

22 

•7  371 
.8  42.4 

•9  47  y 

39 
40 

9 

37289 

52 
52 

9  38534 

55 

55 

0.61  466 

9.98  756 

21 

9 

37341 

9  38  589 

0.61  411 

9  98  753 

20 

41 

9 

37  393 

52 

9  38644 

55 

0  61  356 

9.98750 

19 

51 

4 

42 

9 

37  445 

52 

9  38  699 

55 

0.61  301 

9.98  746 

18 

•  I     51 

0I 

43 

9  37  497 

52 

9  38  754 

55 

0.61  246 

9  98  743 

17 

.2   10.2 

44 
4.S 

9 

37  549 

52 
51 

9  38  808 

54 

55 

0.61  192 

9.98740 

16 

•3    15  3 
.4  20.4 

.6  30. e 

9  45  5 

3 

1.2 

1.6 
2  0 

37600 

9  38  863 
9.38918 

0.61  137 

9  98  737 

15 

46 

9  37  652 

52 

55 

0.61  082 

9  98734 

14 

It 

47 

9  37703 

51 

9.38972 

54 

0.61  028 

9.98  731 

13 

48 

9  37  755 

52 

9  39  027 

55 

0.60973 

9.98  728 

12 

11 

49 
60 

9  37806 

51 

52 

9.39082 

55 
54 

0.60  918 

9.98725 
9.98  722 

II 
10 

9 

37858 

9  39  136 

0 .  60  864 

51 

9 

37909 

51 

9  39  190 

54 

0.60  810 

9  98  719 

9 

0.2 

52 

9 

37960 

51 

9  39245 

55 

0.60  755 

9  98  715 

8 

.2     0.6 

53 

9 

38  on 

^' 

9.39299 

54 

0.60  701 

9  98  712 

7 

0.6 
08 

I.O 

54 
55 

9 

38062 

51 

51 

9  39  353 

54 
54 

0.60  647 

9  98  709 

6 
5 

3     09 
.4     12 

9 

38  113 

9-39  407 

0.60593 

9  98  706 

5^ 

9  38  164 

SI 

9.39461 

54 

0.60539 

9  98  703 

4 

1.2 

■■^^ 

9  38215 

51 

9  39515 

54 

0.60485 

9.98  700 

3 

.7     2.1 
8     2.4 

\i 

5« 

9  38  266 

51 

9  39569 

54 

0  60  43 1 

9.98697 

2 

59 
60_ 

9  383^7 

51 
51 

939623 

54 
54 

0.60377 

9.98694 

I 
0 

9     2.7 

1.8 

9.38368 

9.39677 

0.60323 

9.98690 

L.  Cos. 

d. 

L.  Cotgr. 

C.d. 

L.  Tangr. 

L.  Sin. 

f 

Prop.  Pts.     '   1 

76^                                         1 

40 


TABLE  II 


14 


9_ 
10 
II 

12 
13 

\l 

18 

i9_ 
20 
21 
22 

23 

24^ 

26 
27 
28 
29 
30 
31 
32 
33 
34 

36 

37 
38 
39 
40 

41 

42 

43 
44 

46 

47 
48 

49_ 
50 

51 

52 
53 
il 

II 

60 


L.  Sin.      d. 


38368 
38418 
38469 
38519 
38570 


38  620 
38  670 
38721 

38771 
38821 


38871 
38921 

38971 
39021 
39071 


39  121 
39  170 
39  220 
39270 
39319 


39369 
39418 
39467 

39566 


39615 
39664 

39713 
39  762 
39  811 


39860 
39909 
39958 
40  006 

40055 


40  103 
40  152 
40  200 
40249 
40297 


40346 
40394 
40  442 
40  490 
40538 


40  586 
40634 
40  682 

40730 
40778 


40  825 
40873 

40  921 
4c  968 

41  016 


41  063 
41  III 
41  158 
41  205 
41  252 


41  300 


L.  Cos.     d. 


L.  Tang. 


39677 
39  731 
39785 
39838 
39892 


39  945 

39  999 

40  052 
40  106 
40159 


40  212 
40  266 

40319 
40372 

40425 


40478 
40531 
40584 
40  636 
40  689 


40742 

40795 
40847 
40  900 
40952 


41  005 
41057 
41  109 
41  161 
41  214 


41  266 
41  318 
41  370 
41  422 

41  474 


41  526 
41  578 
41  629 
41  681 
41  733 


41  784 
41  836 
41  887 

41939 
41  990 


42  041 
42093 
42  144 

42  195 
42  246 


42297 
42348 

42399 
42450 
42501 


42552 
42  603 
42653 
42  704 
42755 


9 .  42  805 


L.  Cotg. 


c.d. 


c.d, 


L.  Cotg. 


o .  60  323 
o .  60  269 
0.60  215 
0.60  162 
0.60  108 


0.60  055 
0.60  001 

0.59948 
0.59894 
0.59841 


0.59  788 
0.59734 

0.59  681 
0.59  628 
059  575 


0.59  522 
0.59469 
0.59  416 
0.59364 
0.59311 


0.59258 
o .  59  205 

0.59153 
0.59  100 
0.59048 


0.58995 
0.58943 
0.58891 
0.58839 
0.58  786 


0.58  734 
0.58682 
o .  58  630 
0.58578 
0.58  526 


0.58474 
o .  58  422 
0.58371 
0.58319 
0.58  267 


0.58  216 
0.58  164 
0.58  113 
0.58061 
0.58  010 


0.57959 
0.57907 
0.57856 
0.57805 
o  57  754 


0.57  703 
0.57652 
0.57  601 
0.57550 
o.  57  499 
0.57448 
0.57397 
0.57347 
0.57  296 
0.57245 


0.57195 


L.  Tang. 

75° 


L.  Cos. 


98  690 
98687 
98684 
98681 
98678 


98675 
98671 
98668 
98665 
98662 


98659 
98656 
98652 
98  649 
98646 


98643 
98  640 
98636 
98633 
98630 


98  627 
98  623 
98  620 
98617 
98614 


98  610 
98  607 
98  604 
98  601 
98597 


98594 
98591 
98588 

98584 
98581 


98578 
98574 
98571 
98568 

98565 


98561 
98558 
98555 
98551 

98548 


98545 
98541 
98538 
98535 
98531 


98528 

98525 
98521 
98518 
98515 


98  511 
98508 

98505 
98  501 
98498 


9.98494 


L.  Sin. 


d. 


60 

59 
58 

55 
54 
53 
52 

IL 
50 

49 
48 
47 
_46_ 

45 
44 
43 
42 

41 
40 

39 
38 

35 
34 
33 
32 
31 


25 
24 
23 
22 
21 

20 

19 
18 

17 

16 


Prop.  Pts. 


54 

53 

I 

5i 

5- 

2 

10.8 

10. 

3 

16.2 

15- 

4 

21.6 

21 . 

27.0 

26. 

6 

32.4 

31 

.  7 

37.8 

37- 

.8 

48  6 

42. 

•9 

47 

52 

5- 
10. 

15- 
20. 
26. 

41.6 
46.8 


50 

50 
10.0 
15  o 
20.0 
25.0 
30.0 
350 
40.0 
45  o 


4 

0.4 
o  8 
1.2 
1.6 
2.0 
2.4 
2.8 

36 


48 

47 

I 

4.8 

4 

2 

9.6 

9 

3 

14.4 

14 

4 

19  2 

18. 

S 

24.0 

23 

6 

28.8 

28. 

7 

336 

32 

8 

384 

37 

9 

43  2 

42 

Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


41 


15^ 


L.  Sin. 


0 

I 

2 

3 

I 

I 

10 

II 

12 

13 
14 

;i 

17 

i8 

19 
20 

21 
22 
23 
24 

26 
27 
28 
29 

30 

31 

32 
33 

34 

36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 

J.9 
60 


42  001 
42047 
42093 
42  140 
42  186 


42  232 
42  278 
42324 
42370 
42  416 


42  461 
42507 
42553 
42599 
42644 


42  690 

42735 
42  781 
42  826 
42872 


42917 
42  962 
43008 

43053 
43098 


43  143 
4318S 

43233 
43278 
43323 


43367 
43412 
43  457 
43502 
43  546 


43591 
43  635 
43  680 
43  724 
43  769 


43813 
43857 
43901 
43946 
43990 


44034 


47 
47 
47 
47 
47 

47 
46 

47 
47 
46 

47 
46 

47 
46 

47 
46 
46 

47 
46 
46 

46 
46 
46 
46 
45 
46 
46 
46 
45 
46 

45 
46 
45 
46 
45 

45 
46 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 


L.  Tang. 


C.  d. 


42  805 
42856 
42  906 

42957 
43007 


43  057 
43  108 
43  158 
43  208 
43258 


43308 
43358 
43408 
43458 
43508 


43558 
43607 
43657 
43  707 
43756 


43  806 
43855 
43905 

43  954 

44  004 


44053 
44  102 

44  151 

44  201 
44250 


44299 
44348 
44  397 
44446 
44  495 


44  544 
44592 
44641 
44  690 
44738 


44787 
44836 
44884 

44  933 
44981 


45029 
45078 
45  126 
45  174 
45  222 


45  271 
45319 
45367 
45415 
45463 


45  5" 

45  559 
45  606 
45654 
45  702 


9  45  750 


5» 

50 
51 
50 
50 
51 
50 
50 
50 
50 
50 
50 
50 
50 
50 

49 
50 
50 
49 
50 

49 
50 
49 
50 
49 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 
48 

49 

49 
48 

49 
48 
48 

49 
48 
48 
48 
49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 


L.  Cotg. 


0.57  195 
0.57  144 
0.57094 

0.57043 
0.56993 


o  56943 
o .  56  892 
o .  56  842 
0.56  792 

0.56  742 


o  56  692 
o .  56  642 
0.56  592 

0.56542 
0.56492 


0.56  442 
o  56393 

0.56343 

0.56  293 
0.56  244 


0.56  194 

o  56  145 
0.56  095 
0.56  046 
o  55996 


55  947 
55898 

55849 
55  799 
55  750 


o  55  701 
o  55652 
0.55603 
o  55  554 
o  55505 


o  55456 
0.55  408 

o  55  359 
o  55310 
0.55  262 


o  55213 
0.55  164 
o  55  116 

0.55067 
0.55019 


0.54971 
0.54922 
0.54874 

o .  54  826 
o  54778 


0.54  729 
0.54  681 
0.54633 
o  54585 
o  54  537 


0.54489 
0.54441 
0.54394 
0.54346 
o .  54  298 


0.54250 


L.  Cos.   I  d.  I  L.  Cotg.  c.  d.  L.  Tang. 

740 


L.  Cos. 


9  98  4Q4 
9.98491 
9  98488 

9  98484 
9  98  481 


98477 
98474 
98471 
98467 
98  464 


98  460 
98457 
98453 
98450 
98447 


98443 
98  440 

98436 
98433 
98429 


98  426 
98  422 
98419 

98415 
98  412 


98  409 

98405 
98  402 
98398 
98395 


98391 
98388 

98384 
98381 

98377 


98373 
98370 
98366 
98363 
98359 


98356 
98352 
98349 
98345 
98342 


98338 
98334 
98331 
98327 
98324 


98317 
98313 
98309 
98306 


98  302 
98299 
98295 
98  291 
98288 


9 .  98  284 


L.  Sin, 


d. 


60 

59 
58 

55 
54 
53 
52 
51 
50 
49 
48 
47 
_46_ 

45 
44 
43 
42 
41 
40 
39 
38 

36 

35 
34 
33 
32 
31 
30 
29 
28 
27 
26 


25 
24 
23 
22 
21 

20 

19 
18 

17 
16 


Prop.  Pts. 


5t 

.1 

51 

.2 

10.2 

3 

15  3 

4 

20.4 

•5 

25  5 

6 

30  6 

7 

35-7 

8 

40  8 

9 

45  9 

49 

48 

.1 

4  9 

4 

.2 

9.8 

9 

•3 

14.7 

14 

•4 

19,6 

19 

.  5 

24  5 

24 

.6 

29.4 

28 

7 

34-3 

33 

.8 

39  2 

38 

9 

44  I 

43 

45 

46 

I 

4  7 

4- 

2 

9 

4 

9 

3 
4 

Is 

8 

\l 

5 

11 

5 

23- 

.6 

2 

27 

7 

32 

9 

32 

.8 

37 

6 

36. 

9 

42 

3 

41 

43 

1 

I 

4  51 

2 

9 

0 

3 

13 

5 

4 

18 

0 

5 

22 

5 

6 

27 

0 

■  7 

31 

5 

.8 

36 

0 

9 

40 

5 

Prop.  Pts. 


42 


TABLE  II 


16 


9_ 
10 
II 

12 

13 
14 

15 
i6 

17 
i8 

19 

20 

21 

22 
23 
24 

25 
26 

27 
28 

29 

30 

31 

32 

33 
34 

36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
il 
55 
56 

II 

-59. 
60 


L.  Sin. 


9.44472 
9.44516 

9  44  559 
9.44  602 
9.44646 


45  120 
45  163 
45  206 
45  249 
45  292 


44034 
44078 
44  122 
44  166 
44  210 


44253 
44297 

44341 
44385 
44428 


44  689 

44  733 
44776 
44819 
44  862 


44905 
44948 

44992 
45035 
45077 


45  334 
45  377 
45419 
45  462 
45504 


45  547 
45589 
45632 
45674 
45  716 


45  758 
45  801 
45843 
45885 
45927 


45969 
46  on 
46053 
46095 
46  136 


46  178 
46  220 
46  262 
46303 
46345 


46  386 
46  428 
46  469 
46  511 
46552 


9  46594 
L.  Cos. 


d. 


L.  Tang:. 


45  750 
45  797 
45845 
45892 
45  940 


45987 
46  035 
46  082 
46  130 
46  177 


46  224 
46271 

46319 
46  366 

46413 


46  460 
46507 
46554 
46  601 
46648 


46  694 
46741 
46  788 
46835 
46881 


46  928 

46975 

47  021 
47  068 
47  114 


47  160 
47207 
47253 
47299 
47346 


47392 
47438 
47484 
47530 
47576 


47  622 
47668 

47  714 
47760 
47  806 


47852 
47897 
47  943 
47989 
48035 


48080 
48  126 

48  171 

48217 
48262 


48307 
48353 
48398 
48443 
48489 

48534 

L.  Cotg. 


c.d. 


47 
48 
47 
48 
47 
48 
47 
48 
47 
47 

47 
48 
47 
47 
47 
47 
47 
47 
47 
46 

47 
47 
47 
46 
47 

47 
46 
47 
46 
46 

47 
46 
46 
47 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

45 
46 
46 
46 

45 
46 
45 
46 
45 
45 
46 
43 
45 
46 

45 

c.  d, 


L.  Cotg. 


0.54250 
0.54203 

o  54  155 
0.54  108 
o .  54  060 


o  54013 

0.53965 
0.53918 
0.53870 
0.53823 


0.53  776 
0.53  729 
0.53681 

0.53634 

o- 53  587 


o  53540 
0.53493 
0.53446 
0.53399 
0.53352 


0.53306 
0.53259 
0.53  212 
0.53  165 
0.53  "9 


0.53  072 
0.53025 
0.52979 
0.52  932 
0.52886 


0.52  840 

0.52  793 
0.52  747 
0.52  701 
0.52654 


o .  52  608 
0.52  562 
0.52  516 
0.52  470 
0,52424 


0.52378 
0.52332 
0.52  286 
0.52  240 
0.52  194 


0.52  148 
0.52  103 
0.52057 
0.52  on 
o  51  965 


0.51  920 

0.51  874 

0.51  829 

0.51  783 
0.51  738 


o  51  693 

0.51  647 

0.51  602 
o  51  557 
05^  5" 
0.51  466 

L.  Tang. 

73^ 


L.  Cos. 


98284 
98281 
98277 
98273 
98  270 


98266 

98  262 

98259 

9  98255 

9  98251 

9  98  248 


98  244 
98  240 
98237 
98233 


98  229 
98  226 
98  222 
98218 
98215 


98  211 

98  207 
98  204 
98  200 
98  196 


98  192 
98  189 
98  185 
98  i8i 
98  177 


98  174 
98  170 
98  166 
98  162 
98  159 


98  155 
98  151 

98  147 
08  144 
98  140 


98  136 
98  132 
98  129 
98  125 
98  121 


98  117 
98  113 
98  no 
98  106 
98  102 
98098 
98  094 
98  090 
98087 
98083 
98079 
98075 
98071 
98  067 
98063 


9 .  98  060 
L.  Sin." 


(>0 

It 
1 

55 
54 
53 
52 
11 
50 

49 
48 

47 
i5_ 
45 
44 
43 
42 

41 
40 

39 
38 
37 
Ji 
35 
34 
33 
32 
31 
30 
29. 
28 
27 
26 


25 
24 
23 
22 
21 

20 

19 
18 

17 
16 


15 
14 
13 
12 
II 
To 

9 
8 

7 
6 


Prop.  Pte. 


48 

47 

I 

4.8 

4 

2 

9.6 

9 

3 

14.4 

H 

4 

19  2 

18. 

S 

24.0 

23 

6 

28.8 

28. 

7 

,336 

32 

8 

38  4 

37 

9 

43  2 

42. 

At 

45 

I 

4.6 

4 

2 

9 

2 

9 

3 

13 

8 

IT, 

4 

18 

4 

18. 

5 

23 

0 

22. 

.6 

27 

6 

27 

•  7 

32 

2 

31 

.8 

36 

8 

36. 

9 

41 

4 

40 

44 

43 

I 

4  4 

4 

2 

8.8 

8. 

3 

13  2 

12. 

4 

17.6 

17 

.q 

22.0 

21. 

.6 

26.4 

25 

•  7 

30.8 

30 

.8 

35-2 

34- 

9 

39-6 

38. 

43  ! 

I 
2 
3 
4 

4.2 

8.4 

12.6 

16.8 

21 .0 

6 

25.2 

7 
8 

9 

29  4 
336 
37  ^ 

4 

I 
2 

0.4 
0.8 

3 
4 

S 

1.2 
1.6 
2.0 

.6 

.1 

2.4 
2.8 
32 

9 

3-6 

Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS         43 


17' 


L.  Sin. 


0 

I 

2 

3 
_4 

i 

7 
8 

_9_ 

10 

I 

12 

13 

:i 

17 

i8 

ii. 
20 

21 
22 
23 

24 

25 
26 

27 
28 
29 


31 

32 

33 

36 

37 
38 

39 

40 

41 
42 

43 
44 

46 

47 
48 

50 

51 
52 
53 
il 

60 


46594 
46635 
46  676 
46  717 
46758 


46  8cx) 
46  841 
46882 
46923 
46  964 


47005 

47045 
47  086 
47  127 
47  168 


47209 

47249 
47290 

47330 
47371 


47  411 
47452 
47492 

47  533 
47  573 


47613 
47654 
47694 
47  734 
47  774 


47814 
47854 
47894 
47  934 
47  974 


48  014 
48054 
48  094 
48  133 
48173 


48213 
48252 
48  292 
48332 
48371 


48  411 
48450 
48  490 
48529 
48568 


48607 
48647 
48686 
48  ,725 
48  764 


48803 
48842 
48881 
48  920 
48959 


48998 


41 
41 
41 

41 
42 
41 

41 
41 
41 
41 
40 
41 
41 
41 
41 
40 
41 
40 
41 
40 

41 
40 
41 
40 
40 

41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
39 
40 
40 

39 
40 
40 

39 
40 

39 
40 

39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 


L.  Cos.   I  d. 


L.  Tang. 


c.  d. 


48534 
48579 
48  624 
48669 
48714 


48759 
48804 
48849 
48894 
48939 


48984 
49029 

49073 
49  118 
49  163 


49207 
49252 
49296 
49341 
49385 


49430 
49  474 
49  519 
49563 
49607 


49652 
49  696 
49  740 
49  784 
49828 


49872 
49916 
49960 
50  004 
50  048 


50  092 
50136 
50  180 
50223 
50  267 


503" 

50355 
50398 
50442 
50485 


9  50  529 
9  50572 


50616 
50659 
50703 


50746 
50789 
50833 
50876 
509^9 


50  962 

51  005 
51  048 
51  092 
51  135 


51  178 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
43 
43 
43 
43 
44 
43 
43 


L.  Cotg. 


0.51  466 
0.51  421 
0.51  376 

0-51331 
0.51  286 


0.51  241 
0.51  196 
o  51  151 
o  51  106 
0.51  061 


o  51  010 
o  50971 
0.50  927 

0.50882 
0.50837 


0.50793 
0.50748 

0.50  704 
0.50  659 

0.50615 


0.50  570 
o  50  526 
0.50481 
o  50  437 
0.50393 


o  50  348 
o .  50  304 
o .  50  260 
0.50  216 
0.50  172 


o.  50  128 
o .  50  084 
o .  50  040 

0.49  996 
0.49952 


0.49  908 
o .  49  864 
o .  49  820 
0.49  777 
0.49  733 


o  49  689 

0.49  645 
o  49  602 
0.49  558 
0.49515 


0.49471 
0.49  428 

0.49384 
0.49  341 
0.49  297 


0.49254 
0.49  211 
0.49  167 
0.49  124 
0.49  081 


0.49038 
o  48  995 
o  48  952 
0.48  908 
o  48  865 


0.48822 


L.  Cotg.  c.  d.  L.  Tang. 

72° 


L.  Cos. 


98060 
98  056 
98  052 
98048 
98  044 


98  040 
98  036 
98  032 
98  029 
98025 


98021 
98017 
98013 
98  009 
98  005 


98  001 
97997 

97  993 
97989 
97986 


97982 
97978 
97  974 
97970 
97966 


97962 
97958 
97  954 
97950 
97946 


97942 
97938 
97  934 
97930 
97  926 


97  922 
97918 

97914 
97910 
97906 


97902 
97898 

97894 
97890 


9.97886 
9.97882 
,  97  878 
9.97874 
9.97870 
97866 


97861 
97857 
97853 
97849 
97S45 


97841 
97837 
97833 
97829 
97825 


9782] 


L.  Sin. 


d. 


60 

58 
57 

55 
54 
53 
52 

_5L 
50 

49 

48 
47 

45 
44 
43 
42 
41 
40 

39 
38 

35 
34 
33 
32 

30 

29 
28 

27 
26 


Prop.  Pts. 


45 

I 

4.5 

2 

9.0 

3 

13.5 

4 

18.0 

22.5 

.6 

27.0 

7 

31-5 

8 

36.0 

9 

40.5 

43 

1 

.1 

4  3l 

.2 

8 

6 

•3 

12 

9 

•4 

17 

2 

I 

21 

25 

I 

:l 

30 

I 

34 

4 

9 

38 

7 

41 

I 

4-1 

2 

8.2 

3 

12.3 

4 

16.4 

S 

20.  s 

6 

24.6 

7 

28.7 

.8 

32.8 

9 

36.9 

39 

I 

39 

2 

7-8 

3 

II. 7 

4 

15.6 

5 

19  5 

6 

23  4 

7 

27 -3 

8 

31.2 

9 

35  I 

4 

.1 

0.4 

.2 

0.8 

■3 

1.2 

.4 

1.6 

•5 

2.0 

.6 

2.4 

7 

2.8 

8 

3-2 

9 

36 

4  4 
8.8 
13.2 
17.6 
22.0 
26.4 
30.8 
35  2 
39-6 


42 

4.2 

8.4 
12.6 
16  8 
21 .0 
25.2 
29.4 
33-6 
37-8 


40 

4.0 
8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
32.0 
36.0 


5 

05 
1 .0 

15 

2.0 

2.5 
30 

3-5 
4.0 

4  5 


3 

0.6 

0.9 
1.2 


2.1 

2.4 
2.7 


Prop.  Pts. 


44 


TABLE  II 


18^ 


0 

I 

2 

3 

4 

I 

7 
8 

_9_ 
10 
II 

12 
13 
14 

15 
16 

17 
18 

i9_ 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
35 
31 

36 

37 
38 
J9_ 
40 

41 
42 
43 
44 

46 

47 
48 

49_ 

50 

5^ 
52 
53 
■>! 

]^ 

i9. 
(JO 


L.  Sin. 


9  48998 

9  49037 
49076 

49  115 
49  153 


49  192 
49231 
49269 
49308 
49  347 


9  49385 
9.49424 
9  49462 
9  49  500 
9  49  539 


9  49  577 
9.49615 

9  49654 
49692 

49  730 


49768 
49  806 
49844 
49882 
49920 


49958 
49996 
50034 
50072 
50  no 


50  148 
9.50185 
9.50  223 
9.50  261 
9  50  298 

T  50336" 
9  50  374 
9  50  411 

9  50449 
50486 


9  50523 

9  50  561 

50598 

50635 

50673 


50  710 
50747 
50784 
50  821 
50858 


50  896 

50933 
50970 

51  007 
51  043 


9 
9 
9 
9 

_9_ 

9.51  080 
9  SI  "7 
9  51  154 
9.51  191 
51  227 


9.51  264 


L.  CoSu 


39 
39 
39 
38 
39 

39 
38 
39 
39 
38 

39 
38 
38 
39 
38 

38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

37 
38 
3*? 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
38 
37 
37 
37 
37 
37 
38 

37 
37 
37 
36 
37 
37 
37 
37 
36 
37 


L.  Tang. 


178 
221 
264 
306 
349 


392 
435 
478 
520 
563 


606 
648 
691 

776 
819 
861 

903 
946 


52031 
52073 
52  115 
52157 
52  200 


52  242 
52284 
52  326 
52368 
52410 


52452 
52494 
52536 
52578 
52  620 


52  661 
52703 
52745 
52  787 
52  829 


52  870 
52  912 
52953 
52995 
53037 


53078 
53  120 
53  161 
53  202 
53244 


53285 
53327 
53368 
53409 
53450 


53492 
53  533 
53  574 
53615 
53656 


9  53697 


L.  Cotg.  c.  d 


c.d. 


L.  Cotg. 


o .  48  822 
0.48  779 
0.48  736 
0.48  694 
0.48  651 


0.48608 
0.48  565 
0.48  522 
o .  48  480 

0.48437 


0.48394 
0.48352 

o  48  309 
0.48  266 
o  48  224 


0.48  181 
0.48  139 

o  48  097 
0.48  054 
0.48  012 


0.47969 
0.47927 
0.47885 
0.47843 

0.47  800 


0.47  758 
0.47716 
0.47674 

0.47  632 

0.47590 


0.47548 

0.47  506 

0.47464 

0.47  422 

0.47380 


0.47  339 
0.47297 

0.47255 
0.47213 
0.47  171 


0.47  130 
0.47088 
0.47047 
0.47  005 
0.46963 


0.46  922 
0.46880 
o .  46  839 
0.46  798 
0.46  756 


0.46715 
0.46  673 
o .  46  632 
0.46  591 
0.46  550 


o .  46  508 
0.46  467 
o .  46  426 

0.46  385 
0.46344 


0.46303 


L.  Tang. 

71° 


L.  Cos. 

d. 

9.97821 

60 

9 

97817 

59 

9 

97812 

58 

9 

97808 

57 

9 

97804 

56 

55 

9 

97800 

9 

97796 

S4 

9 

97792 

53 

9 

97788 

52 

9  97  784 

51 
50 

9  97  779 

9-97  775 

49 

9 

97771 

48 

9 

97767 

47 

9 

97763 

46 
45 

9 

97  759 

9 

97  754 

44 

9 

97  750 

43 

9 

97746 

42 

9 

97742 

^ 

41 
40 

9 

97738 

9 

97  734 

39 

9 

97729 

38 

9 

97725 

37 

9 

97721 

36 
3S 

9 

97717 

9 

97713 

34 

9 

97708 

33 

9 

97704 

32 

9 

97700 

31 
30 

9 

97696 

9 

97691 

29 

9 

97687 

28 

9 

97683 

27 

9 

97679 

26 
25 

9 

97674 

9 

97670 

24 

9 

97666 

23 

9 

97662 

22 

9 

97657 

21 
20 

9 

97653 

9 

97649 

19 

9 

97645 

18 

9 

97640 

17 

9 

97636 

16 
15 

9 

97632 

9 

97628 

14 

9 

97623 

13 

9 

97619 

12 

9 

97615 

II 
10 

9 

97  610 

9 

97606 

9 

9 

97602 

8 

9 

97  597 

7 

9 

97  593 

b 

S 

9 

97589 

9 

97584 

4 

9 

97580 

3 

9 

97576 

2 

9 

97571 

I 
0 

9 

97567 

L.  Sin. 

d. 

f 

Prop.  Pte. 


43 

.1 

4  3 

2 

8.6 

3 

12.9 

4 

17.2 

1 

l\i 

7 

30  I 

8 

34-4 

9 

387 

4a 
4.2 


37-8 


41 

41 
8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 

369 


39 

1 

I 

3  9 

2 

7.8 

3 

II 

7 

4 

15 

6 

5 

19 

5 

b 

23 

4 

7 

27 

3 

8 

31 

2 

9 

35 

I 

3^ 

'  1 

I 

3-7| 

.2 

7 

4 

3 

II 

I 

4 

14 

8 

•S 

18 

5 

6 

22 

2 

•  7 

25 

9 

8 

29 

6 

9 

33 

3 

38 

3-8 

76 

II  4 

15  2 
19  o 
22.8 
26.6 

304 
34-2 


36 

3-6 
7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32  4 


5 

I 

05 

2 

1.0 

3 

15 

4 

2.0 

S 

25 

6 

30 

7 

3-5 

8 

4.0 

9 

4-5 

Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS  45 


19^ 


9_ 
10 

12 

13 

\i 

17 
i8 

19 

20 

21 

22 
23 

24 

26 
27 
28 
29 

30 

31 

32 
33 
34 

36 
37 
38 
39 
40 
41 
42 
43 
44 

46 

47 
48 

49_ 

50 

51 
52 
53 

il 

59_ 
60 


L.  Sin. 


264 
301 
338 
374 
411 


447 
484 
520 
557 
593 


629 
666 
702 
738 
774 


811 
847 
883 
919 
955 


51  991 

52  027 
52063 
52099 
52  135 


52  171 
52  207 
52  242 
52  278 
52314 


52350 
52385 
52421 

52456 
52492 


52527 
52563 
52598 
52634 
52  669 


52  705 
52  740 
52  775 
52  811 
52846 


52881 
52  916 

52951 

52  986 

53.0^ 
53056 
53092 

53  126 
53  161 
53  196 


53231 
53  266 
53301 
53336 
53370 


9  53405 


L.  Cos. 


37 
37 
36 
37 
36 

37 
36 
37 
36 
36 

37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 

35 
36 
35 
36 
35 
36 
35 
36 
35 
36 

35 
35 
36 
35 

35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 


L.  Tang. 


53697 
53738 
53  779 
53820 
53861 


c.d. 


53902 
53  943 
53984 
54025 
54065 


54  106 
54147 
54187 
54228 
54269 


54  329 
54350 
54390 
54431 
54471 


54512 
54552 
54  593 
54633 
54673 


54714 
54  754 
54  794 
54835 
54875 


54915 
54  955 
54  995 
55035 
55075 


55  115 
55  155 
55  195 
55235 
55275 


55315 
55  355 
55  395 
55  434 
55  474 


55514 
55  554 
55  593 
55633 
55673 


55712 

55752 
55  791 
55831 
55870 


55910 

55  949 
55989 
56028 

56  067 


56  107 


41 
41 
41 
41 
41 
41 
41 
41 
40 
41 

41 
40 
41 
41 
40 

41 
40 
41 

40 

41 
40 
41 
40 
40 
41 
40 
40 
41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 

39 
40 
40 
40 

39 
40 
40 

39 
40 

39 
40 

39 
40 

39 
40 

39 
39 
40 


L.  Cotg. 


0.46  303 
o .  46  262 
0.46  221 
0,46  180 
0.46  139 


o .  46  098 
0.46057 
0.46  016 
0.45  975 
0.45  935 


0.45  894 
0.45  853 
0.45  813 
0.45  772 
0.45  731 


0.45  691 
0.45  650 
0.45  610 
0.45  569 
0.45  529 


0.45  488 

0.45  448 
0.45  407 

0.45  367 
0.45327 


0.45  286 
0.45  246 
0.45  206 
0.45  165 
0.45  125 


0.45  085 

0.45045 
0.45  005 
0.44965 
0.44925 


0.44885 
0.44845 
o .  44  805 
0.44765 
0.44  725 


o .  44  685 

0.44645 
o .  44  605 
0.44  566 
0.44  526 


o .  44  486 
0.44446 
0.44407 
0.44367 
0.44327 


0.44  288 
o  44  248 
o .  44  209 
0.44  169 
0.44  130 


o .  44  090 
o .  44  05 1 
0.44  on 
o  43972 
o  43  933 


0.43  893 


L.  Cotg.  Ic.  d.  L.  Tang. 

70° 


L.  Cos. 


97567 
97563 
97558 
97  554 
97550 


97  545 
97541 
97536 
97532 
97528 


97523 
97519 
97515 
97510 
97506 


97501 
97  497 
97492 
97488 
97484 


97  479 
97  475 
97470 
97466 
97461 


97  457 
97  453 
97448 

97  444 
97  439 


97  435 
97430 
97426 
97421 
97417 


97412 
97408 
97403 
97  399 
97  394 


97390 
97385 
97381 
97376 
97372 


97367 
97363 
97358 
97  353 
97349 


97  344 
97340 
97  335 
97331 
97326 


97322 
97317 
97312 
97308 
97303 


9.97299 


L.  Sin. 


d. 


60 

59 
58 

57 

55 
54 
53 
52 

_51 
50 

49 
48 
47 

45 
44 
43 
42 

41 
40 

39 
38 

,36 

35 
34 
33 
32 
_3i 
30 

29 
28 

27 
26 


Prop.  Pis. 


4» 

.1 

4.1 

.2 

8.2 

•3 

12.3 

4 

16.4 

•5 

20.5 

6 

24  6 

•7 

28.7 

.8 

32.8 

9 

369 

39 

.1 

3- 

.2 

7- 

•3 

II. 

•  4 

15 

•5 

19 

.6 

23 

•  7 

27. 

.8 

31 

9 

35- 

37 

I 

3-7 

2 

7-4 

•3 

II  .1 

■4 

14.8 

18.5 

.6 

22.2 

•7 

25 -9 

.8 

29.6 

9 

33-3 

3« 

1 

I 

3  51 

.2 

7 

0 

•3 

10 

5 

•4 

14 

0 

.  5 

17 

5 

.6 

21 

0 

•7 

24 

5 

.8 

28  o| 

9 

31 

51 

5 

.1 

05 

.2 

1.0 

•3 

15 

•4 

2.0 

•5 

2.5 

.6 

30 

•  7 

3  5 

.8 

4.0 

•9 

4  5 

40 

4.0 
8.0 

12.0 

16.0 

20.0 
24.0 

28.0 

32  o 

36.0 


36 

36 

7.2 
10.8 
14.4 
18.0 

21  .6 

25.2 

28,8 

32  4 


34 

3  4 
6  8 
10.2 
13  6 
17.0 
20  4 
23  8 
27  2 
30.6 


4 
0.4 

0.8 
1 .2 
1.6 
20 

2.4 
2.8 

36 


Prop.  Pts. 


46 


TABLE  II 


20^ 


0 

I 

2 

3 

I 

7 
8 

10 

II 

12 

13 

'4 

15 
i6 

17 

i8 

i2. 
20 

21 

22 

23 

24 

26 

27 
28 
29 

30 

31 

32 
33 
34 

36 

37 
38 

39 

40 

41 

42 

43 
44 

46 

47 
48 

49 
60 

51 

52 
53 

il 

60 


L.  Sin. 


9  53405 
9  53440 
9  53  475 
9  53509 
9  53  544 


9  53  578 
9  53613 
9  53647 
9  53  682 
9  53  716 


9  53  751 
9  53  785 
9  53819 
9  53854 
9  53  888 


53922 
53  957 
53991 
54025 

54059 


54093 
54  127 
54  161 
54  195 
54229 


54263 
54297 
54331 
54365 
54  399 


54  433 
54466 
54500 
54  534 
54567 


54601 

54635 
54668 
54702 
54  735 


54769 
54  802 
54836 
54869 
54903 


54936 
54969 
55003 
55036 
55069 


55  102 
55  136 
55  169 
55  202 
55235 
55268 
55301 
55  334 
55367 
55400 


55  433 


L.  Cos. 


35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 


L.  Tang. 


56  107 
56  146 
56185 
56  224 
56  264 


56303 
56342 
56381 
56  420 

56459 


56498 

56576 
56615 
56654 


56693 
56732 
56771 
56810 
56849 


56887 
56  926 

56965 
57004 
57042 


57081 
57  120 
57158 
57  197 
57235 


57274 
57312 
57351 
57389 
57428 


57466 
57504 
57  543 
57581 
57619 


57658 
57696 
57  734 
57  772 
57810 


57849 
57887 
57925 
57963 
58  001 


58039 
58077 
58  115 
58153 
58  191 
58  229 
58267 
58304 
58342 
58380 


58418 


d.  L.  Cotg.  c.  d 


c.d. 


L.  Cotg. 


0.43893 
0.43  854 
0.43815 
0.43  776 

0.43  736 


0.43  697 
0.43  658 
0.43  619 
0.43  580 
0.43  541 


0.43  502 

0.43463 
0.43424 

0.43  385 
0.43346 


0.43  307 
0.43  268 
o  43  229 
o  43  190 
o  43  151 


0.43  "3 
0.43074 

o  43035 
o  42  996 
o  42  958 


0.42  919 
o  42  880 
o  42  842 
o .  42  803 
o  42  765 


0.42  726 
o  42688 
o .  42  649 
o  42  611 
0.42  572 


0.42  534 
0.42  496 
0.42457 
0.42  419 
0.42381 


0.42  342 
0.42  304 
o .  42  266 
o  42  228 
o  42  190 


0.42 151 

0.42  113 
o  42075 
o  42037 
0,41  999 


o  41  961 
o  41  923 

0.41  885 

0.41  847 
0.41  809 


o  41  771 
o  41  733 
o  41  696 
o  41  658 
o  41  620 


0.41  582 


L.  Tang. 

69° 


L.  Cos. 


9  97276 
9.97271 
9.97  266 
9.97  262 
9-97  257 


97  206 
97  201 

97  196 
97  192 
97  187 


97299 
97294 
97289 
97285 
97  280 


97252 
97248 

97243 
97238 

97234 


97229 
97224 
97  220 
97215 
97  210 


97  182 
97  178 
97  173 
97  168 
97  163 


97  159 
97  154 
97  149 
97  145 
97  140 


97  135 
97  130 
97  126 
97  121 
97  116 


97  III 

97  107 
97  102 

97097 
97092 


9.97087 
9.97083 
9.97078 

9  97073 
9  97  068 


97063 
97059 
97054 
97049 
97044 


97039 
97035 
97030 
97025 
97  020 


9.97015 


L.  Sin. 


60 

59 
58 
57 

55 
54 
53 
52 
51 
50 
49 
48 
47 

45 
44 
43 
42 
41 
40 
39 
38 
37 

35 
34 
33 
32 
_31 
30 

29 
28 
27 
26 


25 
24 

23 
22 
21 

20 

19 
18 

17 
16 


15 
14 
13 
12 
II 

To 

I 

7 

6 


Prop.  Pte. 


40 

.1 

.2 

Vo 

3 

12.0 

4 

16.0 

c 

20.0 

^ 

24.0 
28.0 

9 

32.0 
36.0 

39 

3  9 

78 

II  7 

156 

19  5 


38 

I 

38 

2 

7.6 

3 

II  4 

4 

15  2 

5 

19  0 

6 

22.8 

7 

26  6 

8 

30  4 

9 

34  2 

37 

3-7 
7-4 
II .  I 
14.8 
18.5 
22.2 
25.9 
29.6 
33-3 


35 

3  5 

7.0 

10.5 

14.0 

17  5 
21 .0 

24  5 
28.0 

31  5 


34 

33 

I 

3-4 

3. 

2 

6.8 

6. 

3 

10.2 

9 

4 

13  6 

13. 

5 

17  0 

16. 

6 

20  4 

19. 

7 

23.8 

23- 

8 

27  2 

26. 

9 

30.6 

29. 

5 

I 

05 

2 

I.O 

3 

15 

•4 

2.0 

.5 

2    S 

.6 

30 

•7 

3  5 

.8 

4.0 

9 

4  5 

4 

0.4 
0.8 
1.2 
1.6 
2  0 
2.4 
28 

^l 
36 


Prop,  Pt8. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


47 


21 


9_ 
10 

12 

13 

;i 

17 
i8 

19 

20 

21 
22 
23 

24 

26 

27 
28 

29 

30 

31 
32 

33 
34 

36 

37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 

50 

51 
52 

53 
54 

55 
56 
57 
58 
59 
60 


L.  Sin. 


9  55  433 
9  55466 

9  55  499 
9  55532 
9  55  564 


9  55  597 
9  55  630 
9  55663 
9  55  695 
9  55  728 


55  761 
55  793 

55  858 
55  891 


55923 
55  956 
55988 
56021 
56053 


56085 
56  118 

56150 
56  182 
56215 


56247 
56279 
563" 
56343 
56375 


56  408 
56440 
56472 
56504 
56536 
56568 

56599 
56631 
56663 
56695 


56727 
56759 
56  790 
56822 
56854 


56886 
56917 
56949 
56980 
57012 


57044 
57075 
57  107 
57138 
57  169 


9.57201 
9  57232 
9  57264 
9  57  295 
9  57  326 


9-57  358 


L.  Cos. 


L.  Tang«t€.  i>  L.  Cotg 


58418 
58455 
58493 
58531 
58569 


58606 
58644 
58681 
58719 
58757 


58794 
58832 
58869 
58907 
58944 


58981 
59019 
59056 
59094 
59  131 


59  168 
59205 
59243 
59  280 
59317 


59  354 
59391 
59429 
59466 

59503 


59540 
59  577 
59614 
59651 
59688 


59725 
59762 
59  799 
59835 
59872 


59909 
59946 
59983 
60  019 
60  056 


60093 
60  130 
60  166 
60  203 
60  240 


60  276 
60  313 

60349 
60386 
60422 


60459 
60495 
60532 
60568 
60  605 


60  641 


d.     L.  Cotg. 


37 

38 
38 
38 
37 
38 
37 
38 
38 
37 
38 
37 
38 
37 
37 
38 
37 
38 
37 
37 
37 
38 
37 
37 
37 
37 
38 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
37 
37 
36 
37 
37 
37 
36 
37 
37 
36 

37 
36 
37 
36 
37 
36 
37 
36 
37 
36 


41  582 
41  545 
41  507 
41  469 

41  431 


41  394 
41  356 
41  319 
41  281 
41  243 


41  206 
41  168 

41  131 

41093 
41  056 


41  019 
40  981 

40944 
40  906 
.  40  869 


40832 
40795 
40757 
40  720 
40683 


40  646 
40  609 
40571 
40534 
40497 


40  460 

40423 
40  386 

40349 
40312 


40275 
40238 
40  201 
40  165 
40  128 


40  091 
40054 
40  017 
39981 
39  944 


L.  Cos. 


39907 
39870 
39834 
39  797 
39  760 


39  724 
39687 
39651 
39614 
39578 


39  541 
39505 
39468 

39432 
39  395 


9.97015 
9.97  010 
9.97005 
9  97  001 
9  96  996 


9  96  991 
9  96  986 


9 .  q6  942 
9.96937 


96932 
96927 
96  922 


39  359 


c.  d.  L.  Tang. 

68° 


96981 
96  976 
96971 


96  966 
96  962 
96957 
96952 
96947 


96917 
96  912 
96907 
96  903 
96898 


96893 
96888 
96883 
96878 
96873 


96868 
96863 
96858 

96853 
96848 


d. 


Q6843 
96838 
96833 
96828 
96823 


96818 
96813 
96808 
96803 
96798 


96  793 
96788 

96783 
96778 
96  772 


96  767 
96  762 
96757 
96  752 
96  747 


96  742 
96737 
96  732 
96  727 
96  722 


9.96717 


L.  Sin. 


60 

59 
58 

55 
54 
53 
52 
_5i_ 
50 
49 
48 
47 

45 
44 
43 
42 
41 
40 

39 
38 

_36 

35 
34 
33 
32 
31 


Prop.  Pts. 


25 
24 
23 
22 
21 
20^ 

19 

18 

17 
16 


10 

I 

7 
6 


38 

I 

3  8 

2 

76 

3 

11  4 

4 

15  2 

5 

19  0 

6 

22  8 

7 

26  6 

8 

30  4 

9 

342 

37 

3  7 
1  \ 
II .  I 
14  8 
185 
22  2 

25  9 
29  6 

33  3 


36 

33 

I 

36 

3 

2 

72 

6. 

3 

10  8 

9 

4 

S 

18  0 

\l 

6 

21.6 

19 

•7 

25.2 

23 

.8 

28.8 

26 

9 

32.4 

29 

33 

64 
96 

12    8 

16  o 

19   2 
22.4 

25  6 
28.8 


31 

.2 

6.2 

3 

•4 

i 

9  3 
12.4 

•  7 
.8 

9 

21.7 
24.8 
27.9 

5 

I 

05 

2 

I.O 

3 

15 

4 

20 

5 

2.5 

6 

30 

7 

3-5 

8 

4.0 

9 

4  5 

6 

0.6 
1.2 
I 

2.4 

3-6 
4.2 
4.8 
5-4 


4 

0.4 
O. 
I    2 

1.6 
2.0 

24 
2.8 

36 


Prop.  Pts. 


48 


TABLE  II 


22' 


0 

I 

2 

3 
_4 

I 

7 
8 

_9_ 

10 

II 

12 

13 

ii_ 

;i 

17 
i8 

19 

20 

21 

22 
23 

24 

26 
27 
28 
29 

30 

31 

32 
33 

36 

38 

39 
40 

41 
42 
43 
44 

46 

47 
48 

49 
50 

51 

52 
53 
54. 

II 
II 

59 
60 


L.  Sin. 


57358 
57389 
57420 

57451 
57482 


57514 
57  545 
57576 
57607 
57638 


57669 
57700 

57731 
57762 

57  793 


57824 
57855 
57885 
57916 
57  947 


57978 
58008 
58039 
58  070 
58  loi 


58  131 
58  162 
58  192 
58223 
58253 


58284 
58314 
58345 
58375 
58  406 


58436 
58467 
58497 
58527 
58557 


58588 
58618 
58648 
58678 
58709 
58  739 
58769 
58799 
58829 
58859 


58889 
58919 
58949 
58979 
59009 


59039 
59069 
59098 
59128 
59  158 
59188 

L.  Cos. 


31 
31 

31 
31 
32 
31 
31 

3» 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
30 
31 
31 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 
35^ 
30 
30 
30 
31 
30 
30 
30 
31 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
29 


L.  Tangr. 


9.60  641 
9.60  677 
9.60  714 
9  60  750 
9 . 60  786 
9 .  60  823 
9  60  859 
9  60  895 
9.60  931 
9  60  967 


9.6 

9.6 

^i 
9.6 


9.6 

9.6 


9.6 

9.6 
9.6 


9.6 
9.6 

9.6 


9.6 
9.6 
9.6 


96 
9.6 
9.6 


004 
040 
076 
112 
148 


184 
220 
256 
292 
328 


364 
400 

436 
472 
508 


544 
579 
615 
651 
687 


722 
758 
794 
830 
865 


901 

936 
972 


9 ,  62  008 
9.62043 


9 . 62  079 
9.62  114 
9.62  150 
9.62  185 
9.62  221 


9.62  256. 
9 . 62  292 
9.62327 
9 .  62  362 
9.62  398 


9  62  433 
9 .  62  468 
9.62  504 
9  62539 
9  62  574 


c.  d.  L.  Cotg. 


9.62  609 
9.62  645 
9 .  62  680 
9.62  715 
9.62  750 
9.62  785 

L.  Cotg. 


36 
37 
36 
36 
37 
36 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 

35 
36 
36 
36 
35 
36 

35 
36 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
36 
35 
35 
35 
36 
35 
35 
35 
35 

c.  d. 


0.39359 
0.39323 
0.39  286 
0.39  250 
0.39214 


0.39  177 
0.39  141 
0.39  105 
o .  39  069 
0.39033 


o  38  996 
o .  38  960 

0.38924 
0.38888 

0.38852 


0.38816 
0.38  780 

0.38  744 
0.38  708 
0.38672 


0.38636 
o .  38  600 
o .  38  564 
0.38528 
0.38492 


0.38456 
0.38  421 
0.38385 
0.38349 
0.38313 


0.38278 
0.38  242 
o .  38  206 
0.38  170 

0.38135 


0.38099 

o .  38  064 

0.38028 
0.37992 
0.37957 


0.37921 
0.37886 
0.37850 
0.37815 

o  37  779 


0.37  744 
0.37708 
0.37673 
0.37  638 
0.37  602 


o  37567 
0.37532 
0.37496 
0.37461 
o  37  426 


0.37391 
o  37  355 
o  37320 
0.37285 
o  37250 
o  37215 
li.  Tang. 

67° 


L.  Cos, 


96717 
96  711 

96  706 
96  701 
96  696 


96  691 
96686 
96681 
96  676 
96  670 


96  665 
96  660 
96655 
96  650 
96645 


96  640 
96634 
96  629 
96  624 
96  619 


96  614 
96608 
96603 
96598 
96593 


96588 
96  582 
96577 
96572 
96567 


96  562 
96556 
96551 
96546 

96541 


96535 
96530 
96525 
96  520 

96514 


96509 
96504 
96  498 

96493 
96488 


96483 
96477 
96472 
96467 
96  461 


96456 
96451 
96445 
96  440 

9^35. 
96  429 
96424 
96419 

96413 
96  408 

9  96403 

L.  Sin. 


60 

59 
58 

1 

55 

54 

53 

52 

S]_ 

50 

49 

48 

47 

45 
44 
43 
42 
41 
40 

39 
38 

_3i 
35 
34 
33 
32 
31 


Prop.  Pts. 


35 

1 

.1 

3-71 

.2 

7 

4 

•3 

II 

I 

4 

14 

8 

.5 

18 

5 

6 

22 

2 

7 

25 

9 

.8 

29 

6 

9 

33 

3 

36 

36 
7.2 
10.8 
14.4 
18.0 

21.6 

25.2 
28.8 

32  4 


35 

3  5 
7.0 

10.5 
14.0 

17  5 
21  o 

24  5 
28.0 

31  5 


32 

I 

32 

2 

6.4 

3 

9.6 

4 

12.8 

5 

16.0 

.6 

19.2 

7 

22.4 

.8 

25.6 
28.8 

9 

30 

29 

I 

2 

3.0 
6.0 

2 

3 

4 

9.0 
12.0 

II . 

I 

15  0 
18.0 

14 
17 

I 
9 

21.0 

20 

24.0 
27  0 

23 
26 

6 

0.6 
1.2 

1.8 
24 

36 
4.2 
4.8 
5  4 


31 

3^ 

6.2 

9  3 
12.4 

\ll 

21.7 
24.8 
27.9 


5 

05 


I 
I 
2.0 

2  5 

30 

3  5 
4.0 

4  5 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 

49 

23°                                        1 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

9  59  188 

9.62785 

0.37215 

9.96403 

6 

■fiO" 

I 

9  59  218 

9 

62820 

35 
35 
36 
35 

0.37  180 

9 

96397 

59 

2 

9 

59247 

9 

62855 

0.37  145 

9 

96392 

5 

58 

36 

3 

7 

10 

35 

6     3-5 
2     7.C 
8   10.5 
4.   14.0 

3 

9 

59277 

30 
30 
29 

9 

62890 

0.37  no 

9 

96  387 

57 

_4 

9 

59307 

9 

62  926 

0.37074 

9 

96381 

5 
6 

56 

55 

2 
3 

9 

59336 

9 

62  961 

0.37039 

9 

96376 

6 

9 

59366 

9 

62  996 

0  37004 

9 

96370 

54 

\/\ 

7 

9 

59396 

9 

63031 

0.36969 

9 

96365 

53 

5 

t8 

0   17.1; 

8 

9 

59425 

9 

63  066 

35 
34 

35 

0.36934 

9 

96360 

5 
6 
5 

52 

6 

21 

6   21  0 

9 
10 

9 

59  455 

29 

9 

63  lOI 

0 .  36  S99 

9 

96354 

51 
50 

7 
8 

li. 

2   24.5 
8   28.0 

9 

59484 

9 

63135 

0.36865 

9 

96349 

II 

9 

59514 

9 

63  170 

0 .  36  830 

9 

96343 
96338 

49 

9 

32. 

4  31  5 

12 

9 

59  543 

9 

63205 

0.36  795 

9 

48 

1 

i.S 

9 

59  573 

9 

63  240 

35 
35 

0.36  760 

9 

96333 

6 
5 
6 

47 

1 

14 
15 

9 

59602 

30 

9 

63275 

0.36725 

9 

96327 

46 
45 

.1 

34 

9 

59632 

9 

63310 

0.36690 

9 

96322 

16 

9 

59  661 

9 

63345 

■lA 

0.36655 

9 

96316 

44 

.2 

6.8 

17 

9 

59690 

29 

9 

63379 

0.36  621 

9 

96  311 

5 
6 

43 

•3 

[O  2 

18 

9 

59  720 

30 

9 

63414 

0.36586 

9 

96305 

42 

4 

[36 

19 
20 

9 

59  749 

29 

9 

63449 

35 

0.36551 

9 

96300 

5 
6 

41 
40 

i: 

17.0 

20.4 
238 

30.6 

9 

59778 

9 

63484 

0.36516 

9 

96294 

21 

9 

59808 

9 

63519 

0.36481 

9 

96289 

5 

39 

22 

9 

59^7 

29 

9 

63  553 

0.36447 

9 

96284 

5 
6 

38 

23 

9 

59866 

29 

9 

35 

0.36412 

9 

96278 

37 

■y  1  . 

24 

2S 

9 

59895 

29 

29 

9 

63623 

34 

0.36377 

9 

96273 

5 
6 

36 

35 

1 

9 

59924 

9 

63657 

0.36343 

9 

96267 

30 

29 

2b 

9 

59  954 

9 

63692 

0.36  308 

9 

96  262 

5 

34 

27 

9 

59983 

29 

9 

63  726 

34 

0.36274 

9 

96256 

33 

i. 

0     5.8 
0     8.7 
0    II. 6 
0  14.  ■; 

28 

9  60012 

29 

9 

63  761 

0.36239 

9 

96251 

5 

32 

29 

30 

9  60  041 

29 

29 

9 

63796 

34 

0 .  36  204 

9 

96245 

5 
6 

31 
30 

i 
4 

9 

12. 
15 . 

9 . 60  070 

9 

63830 

0.36  170 

9 

96  240 

31 

9 

60099 

9 

63  865 

0.36  135 

9 

96234 

29 

18. 

0  17  4 

32 

9 

60128 

9 

63899 

0.36  lOI 

9 

96  229 

5 
6 

28 

I 

21 . 

0  20  3 
0  23  2 

33 

9 

60157 

29 

9 

63934 

0.36066 

9 

96  223 
96  218 

27 

?./[ 

34 
35 

9 

60186 

29 

29 

9 

63968 

35 

0.36032 

9 

5 
6 

26 
25 

9 

27. 

0  26  I 

9 

60215 

9 

64  003 

0.35997 

9 

96  212 

3^^ 

9 

60244 

29 

9 

64037 

0.35963 

9 

96  207 

5 
6 

24 

1 

37 

9 

60273 

29 

9 

64  072 

35 

0  35  928 

9 

96  201 

23 

38 

3« 

9 

60302 

29 

9 

64  106 

34 

0.35894 

9 

96  196 

5 
6 
5 

22 

T 

2  8 

39 
40 

9 

60331 

29 
28 

9 

64  140 

35 

0.35860 

9 

96  190 

21 
20 

.2 

3 

56 
8.4 

9 

60359 

9 

64  175 

0  35825 

9 

96  185 

41 

9 

60388 

29 

9 

64  209 

34 

0.35  791 

9 

96179 

19 

.4 

II  .2 

42 

9 

60417 

29 

9 

64243 
64278 

0.35  757 

9 

96  174 

5 

18 

s 

14  0 

43 

9 

60  446 

29 
28 
29 

9 

0.35  722 

9 

96168 

17 

6 

16  8 

44 

9 

60474 

9 

64312 

34 

0.35  688 

9 

96  162 

5 

16 
15 

•  7 
.8 

196 
22  4 

45 

9 

60503 

9  64  346 

o- 35  654 

9 

96  157 

46 

9 

60  532 

29 

9  64  381 

0.35619 

9 

96  151 

14 

•9    - 

25  2 

47 

9 

60  561 

29 
28 

9.64415 

o- 35  585 

9 

96  146 

5 
6 

13 

1 

48 

9 

60589 

9 

64449 

34 

0.35551 

9 

96  140 

12 

1 

49 
50 

9 

60618 

29 

28 

9 

64483 

34 
34 

0.35517 

9 

96135 

5 
6 

II 
10 

6 
0 

5 

6     0.5 

9 

60  646 

9 

64517 

0  35483 

9 

96  129 

51 

9 

60675 

29 

9 

64552 

35 

0.35448 

9 

96  123 

9 

2 

I 

2       I.O 

52 

9 

60704 

29 

9 

64586 

34 

0.35414 

9 

96  118 

5 

8 

3 

I 

8     1.5 

53 

9 

60732 

9 

64  620 

34 

0.35380 

9 

96  112 

7 

4 

2 

^     2.0 

54 

55 

9 

60  761 

29 
28 

9 

64654 

34 
34 

0.35  346 

9 

96  107 

5 
6 

6 

5 

3  < 
3 

5     3.0 

9 

60789 

9 

64688 

0.35312 

9 

96  lOI 

56 

9 

60818 

29 

9 

64  722 

34 

0.35  278 

9 

96095 

4 

7 

4 

I     3  5 
i    40 

57 

9 

60846 

9 

64756 

34 

0.35  244 

9 

96  090 

5 

3 

8 

4^ 

5« 

9 

60875 

29 

9 

64790 

34 

0.35  210 

9 

96084 

2 

9 

b-' 

^     4  5 

59 
60_ 

9 

60903 

28 

9 

64824 

34 
34 

0.35  176 

9 

96079 

5 
6 

I 
0 

9 

60931 

9 

64858 

0.35  142 

9 

96073 

L.  Cos. 

d. 

L.  Cotgr. 

c!T 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pt8. 

66^ 



50 


TABLE  II 


24' 


2 

3 

1 

7 
8 

9 
10 
II 

12 

13 

\l 

17 
i8 

19 

20 

21 

22 
23 

24 

25 
26 

27 
28 

29 

30 

31 
32 

33 
J4 

35 
3^ 

37 
38 

39 

40 

41 

42 
43 
44 

46 

47 
48 

49 

60 

51 

52 
53 
ii 

II 

57 
58 

Ji. 
60 


L.  Sin. 


6093: 
60  960 
60988 


016 
045. 
073 

lOI 

129 

158 
186 


214 

242 
270 

298 
326 


354 
382 
411 
438 
466 


494 
522 

578 
606 


634 
662 
689 
717 
745 


773 
800 
828 
856 
883 


911 

939 
966 

994 
62  021 


62  049 
62  076 
62  104 
62  131 
62  159 


d. 


62  186 
62  214 
62  241 
62  268 
62  296 


62323 
62  350 
62377 
62  405 
62432 


62459 
62486 

62513 
62541 
62568 


62595 


L.  Cos. 


29 
27 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
27 
28 
28 
27 
28 
27 
28 

37 

28 
27 
28 


L.  Tang. 


c.d. 


64858 
.  64  892 
.  64  926 
.64  960 
64  994 


65  028 
65  062 
65  096 


197 
231 
265 
.  299 
65333 


65  366 
9 .  65  400 

9  65  434 
9.65467 

9  65  501 


65535 
65568 
65  602 
65  636 
65  669 


65  703 
65  736 
65  770 
65  803 
65837 


130 
164 


65870 
65  904 
65937 

65  971 

66  004 


66038 
66071 
66  104 
66  138 
66  171 


66  204 
66238 
66  271 
66  304 
66337 


9.66371 
9 .  66  404 
9.66437 
9 .  66  470 
9  66  503 


9  66537 
9  66  570 
9 .  66  603 
9 .  66  636 
9  66  669 


9.66  702 
9  66735 
9  66  768 
9.66  801 
9.66834 


66867 


L.  Cotg. 


L.  Cotg. 


0.35  142 
0.35  108 

o  35074 
0.35  040 
0.35  006 


o  34  972 
0.34938 
0.34904 
0.34870 
o  34  836 


o  34  803 
0.34  769 

0.34735 
0.34  701 
0.34667 


0.34634 
o .  34  600 
0.34566 
0.34533 
0.34499 


0.34465 
0.34432 
0.34398 
0.34364 
0.34331 


0.34297 
o .  34  264 
0.34230 
0.34  197 
0.34  163 


0.34  130 
0.34096 
o .  34  063 
o .  34  029 
0.33996 


0.33  962 

0.33929 
0:33896 
0.33  862 
0.33829 


0.33  796 
0.33  762 

0.33  729 
0.33696 
o  33663 


0.33  629 

0.33  596 
0.33563 
033  530 
0.33497 


0.33463 
0.33430 
0.33397 
0.33364 
0.33  331 


c.d. 


0.33  298 
o  33  265 
o  33  232 
0.33  199 
0.33  166 


0.33  133 


L.  Tang. 

65° 


L.  Cos, 


9.96073 
9 .  96  067 
9 .  96  062 
9.96  056 
9 ,  96  050 


9  96  045 
9  96  039 
9  96  034 
9  96  028 
9  96  022 


9.96  017 
9.96  on 
9 .  96  005 
9.96  000 
9  95  994 


9.95988 
9.95982 
9  95  977 
9  95971 
9  95965 


9  95  960 
9  95  954 
9  95948 
9  95942 
9  95  937 


9  95931 
9  95925 
9.95920 

9  95  914 
9  95908 


9  95  902 
9  95897 
9  95  891 
9.95885 

9  95879 


9-95873 
9  95  868 
9.95  862 
9  95856 
9  95850 


9-95  844 
9  95839 
9  95833 
9.95827 
9.95821 


9  95815 
9.95  810 
9.95  804 
9  95  798 
9  95  792 


9  95  786 
9  95  780 
9  95  775 
9  95  769 
9  95  763 


9  95  757 
9  95  751 
9  95  745 
9  95  739 
9  95  733 


9  95  728 


L.  Sin, 


d. 


60 

59 
58 

1 

55 
54 
53 

52 

iL 
50 

49 
48 
47 

45 
44 
43 
42 
41 
40 

It 
1 

35 
34 
33 
32 
_3i 
30 
29 
28 

27 
26 


Prop.  Pte. 


34 

33 

I 

^A 

J. 

2 

6.8 

6. 

3 

10   2 

9 

4 

13  6 

13 

5 

17.0 

16 

6 

20,4 

19 

7 

23.8 

23 

8 

27.2 

26 

9 

30.6 

29 

29 

,  I 

2.9 

.2 

5.8 

•3 

8.7 

.4 

II. 6 

•5 

145 

.6 

17  4 

•7 

20.3 

.8 

23.2 

•9 

26.1 

a8 

.1 

2.8 

.2 

5  6 

.3 

8.4 

•4 

II. 2 

14.0 

6 

16.8 

i 

19.6 

22.4 

•9 

25.2 

37 

.1 

2.7 

.2 

5  4 

•3 

8. 1 

•4 

10.8 

•S 

13  5 

.6 

16.2 

i 

18.9 

21  6 

9 

24  3 

6 

.1 

0.6 

.2 

12 

•3 

1.8 

4 

2.4 

.  ^ 

3  c 

6 

3-6 

7 

4.2 

8 

4-8 

9 

5  4 

5 

05 
1 .0 

15 
2.0 

2.5 
30 

3  5 
4.0 

4-5 


Prop.  Pts, 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


51 


25" 


L.  Siu. 


_9_ 
10 
II 

12 

13 

!i 

i8 
19 
20 

21 
22 

23 
24 

26 
27 
28 
29 

30 

31 
32 
33 
34 

36 
37 
3« 
39 
40 
41 
42 
43 
44 

46 

47 
48 

j49 

50 

51 

52 

53 

H 

55 

56 

II 

GO 


62595 
62  622 
62  649 
62  676 
62  703 


62  730 
62  757 
62  784 
62  811 
62838 


62865 
62  892 
62918 
62945 
62  972 


62  999 

63  026 
63  052 
63079 
63  106 


63  133 

63213 
63239 


63  266 
63  292 
633^9 
63345 
63372 


63398 
63425 
63451 
63478 
63  504 


63531 

63557 

63  610 
63636 


63  662 
63  689 
63  715 
63  741 
63  767 


63  794 
63820 
63  846 
63872 
63898 


63924 
63950 
63976 
64  002 
64  028 


64054 
64  080 
64 106 
64 132 
64 158 


64 184 


L.  Cos. 


d. 


L.  Tang. 


66867 
66  900 
66933 
66966 
66  999 


67032 
67065 
67098 

67  131 
67  163 


67  196 
67  229 
67  262 
67295 
67327 
67360 

67393 
67426 

67458 
67491 


67524 
67556 
67589 
67  622 

67654 


67687 
67719 
67752 
67785 
67817 


67850 
67882 
67915 
67947 
67  980 


68012 
68  044 
68077 
68  109 
68  142 


68  174 
68206 
68239 
68271 
68303 


68336 
68368 
68  400 
68432 
68465 


68497 
68529 
68561 

68593 
68626 


68658 
68690 
68  722 

68754 
68  786 


68818 


Cotgr. 


L.  Cotg. 


0-33  133 
0.33  100 
0.33067 

0.33034 
0.33001 


o  32  968 

0.32935 
0.32  902 
o .  32  869 

0.32837 


o .  32  804 

0.32  771 
0.32  738 
0.32  705 
0.32673 


o .  32  640 
0.32  607 

0.32  574 
0.32  542 
0.32  509 


0.32476 
0.32444 
0.32  4n 
0.32378 
0.32346 


0.32313 
0.32  281 
0.32  248 
0.32  215 
0.32  183 


0.32  150 
0.32  118 
o .  32  085 
0.32  053 
o .  32  020 


0.31  988 
0.31  956 
o  31  923 
0.31  891 

0.31  858 


0.31  826 

o  31  794 
0.31  761 
0.31  729 
0.31  697 


0.31  664 
0.31  632 
0.31  600 
0.31  568 
o. 31535 


0.31  503 
0.31  471 

o  31  439 
0.31  407 
0.31  374 


342 
310 
278 
246 
214 


0.31  182 


c.  d.  L.  Tang. 

64^ 


L.  Cos. 


9.95698 
9.95692 
9.95  686 
9.95  680 
9  95674 


95  728 
95  722 
95  716 
95  710 
95  704 


95668 
95663 
95657 
95651 
95645 


95639 
95633 
95627 
95  621 
95615 


95609 
95603 
95  597 
95591 
95585 


95  579 
95  573 
95567 
95561 
95  555 


95  549 
95  543 
95  537 
95531 
95525 


95519 
95513 
95507 
95500 
95  494 


95488 
95482 
95476 
95470 
95464 


95458 
95452 
95446 
95440 
95  434 


95427 
95421 
95415 
95409 
95403 


95  397 
95391 
95384 
95378 
95372 


9.95366 


d. 


L.  Sin.   I  d. 


60 

59 
58 
57 

55 
54 
53 
52 
_51 
50 

49 
48 
47 
_46_ 

45 
44 
43 
42 
41 
40 

39 
38 

_3^ 
35 
34 
33 
32 
31 


25 
24 
23 
22 
21 

w 

19 

17 
16 


15 
14 
13 

12 
II 

To 

i 

7 
6 


Prop.  Pts. 


33 

I 

3  3 

2 

6.6 

3 

9.9 

4 

13.2 

16.5 

6 

19.8 

7 

8 

26.4 

9 

29.7 

33 

6.4 

9.6 

12.8 

16.0 
19.2 

22.4 
25.6 
28.8 


27 

2.7 

i;t 

10.8 
16.2 

.8.9 

21.6 
24-3 


a6 

2.6 

7.8 
10.4 

lie 

18  2 
20  8 
23  4 


7 

0.7 
1-4 

2.1 
2.8 

3  5 

4  2 
4-9 

l^ 
6.3 


.1 

6 
0.6 

.2 

12 

•3 

I  8 

•4 

2  4 

it 

i 

t:i 

•9 

5-4 

5 

0.5 
i.o 

1-5 
2  o 

2.5 
30 

3-5 
4.0 

4-5 


Prop.  Pts. 


52 


TABLE  II 


26' 


2 

3 
± 

I 

7 
8 

10 

II 

12 

13 
14_ 

IS 

i6 

17 

i8 

II. 
20 

21 

22 
23 

26 
27 
28 

30 

31 

32 
33 
Ji 

39 
40 

41 

42 

43 
44 

46 

47 
48 

49 

50 

51 
52 
53 

54. 

_59 
GO 


L.  Sill. 


64  184 
64  210 
64236 
64  262 
64288 


64313 
64339 
64365 
64391 
64417 


9  64442 
9  64  468 

9  64  494 
9  64519 

9  64545 


964  571 
9.64596 
9 .  64  622 
9.64647 
9.64673 


9 .  64  698 
9.64724 
9.64  749 

9  64775 
9 .  64  800 


9 .  64  826 
9.64  851 
9,64877 
9 .  64  902 
9  64927 


9  64953 
9.64978 
9.65003 
9 .  65  029 
9  65054 


9.65  079 
9  65  104 
9.65  130 

9  65  155 
9.65  180 


9 .  65  205 
9.65  230 
9  65255 
9.65  281 
9.65  306 


9  65331 
9  65356 
9  65381 
9  65  406 

9  65431 


d. 


9  65456 
9.65  481 
9.65  506 
9  65531 
9  65556 


9  65  580 
9  65  605 
9  65  630 
9  65655 
9  65  680 


9  65705 


L.  Cos. 


26 
26 
26 
26 
25 
26 
26 
26 
26 
25 
26 
26 
25 
26 
26 

25 
26 

25 

26 

25 

26 
25 
26 
25 
26 

25 
26 
25 
25 
26 

25 
25 
26 

25 
25 

25 
26 
25 
25 
25 
25 
25 
26 
25 
25 

25 

25 
25 
25 
25 

25 

25 
25 
25 
24 
25 
25 

25 

25 
25 


L.  Tang. 


c.d. 


d. 


9.68818 
9.68850 
9.68882 
9.68  914 
9 .  68  946 


9.68978 
9.69  010 
9 .  69  042 
9.69074 
9.69  106 


9.69  138 
9.69  170 
9 .  69  202 
9  69234 
9 .  69  266 


9 .  69  298 
9.69329 
9.69361 

9  69  393 
9.69425 


9  69  457 
9 .  69  488 
9.69  520 
9  69  552 
9.69584 


9  69  615 
9.69647 
9.69679 
9.69  710 
9.69742 


9.69  774 
9 ,  69  805 
9.69837 
9.69868 
9 .  69  900 


9.69932 
9.69963 

9  69995 
9  70  026 
9 .  70  058 


9 .  70  089 
9  70  121 
9.70  152 
9.70  184 
9  70  2 1 5 


9.70247 
9.70  278 
9.70309 

9  70341 
9.70372 


9.70404 

9  70435 
9  70  466 
9  70498 
9  70529 


9 . 70  560 
9.70592 
9  70  623 
9.70654 
9 .  70  685 


9.70717 


32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

31 
32 
32 

32 
32 

31 
32 
32 
32 
31 
32 
32 
31 
32 
32 

31 
32 
31 
32 
32 

31 
32 

3» 
32 
31 
32 
31 
32 
31 
32 

31 
31 
32 
31 
32 

31 
31 
32 
31 
31 
32 
31 
31 
31 
32 


L.  Cotg. 


0.31 
0.31 


[82 

-  % 
0.31  118 

o  31  086 

o  3»  054 


o  31  022 
o .  30  990 
0.30  958 
o .  30  926 

0.30894 


o .  30  862 
o .  30  830 
0.30  798 
0.30  766 
o  30  734 


0.30  702 
0.30  671 
o .  30  639 
0.30  607 
o  30575 


0.30543 

0.30  512 
o .  30  480 

0.30448 

o  30  416 


0.30  385 

o.  30  353 
0.30  321 
o .  30  290 
0.30258 


o .  30  226 

0.30  195 

0.30  163 
0.30  132 
0.30  100 


L.  Cos. 


o .  30  068 
o  300^7 
o .  30  005 

0.29974 

o .  29  942 


0.29  911 
0.29  879 
o .  29  848 
0.29  816 
0.29  785 


0.29753 

0.29  722 
0.29  691 
0.29  659 
0.29  628 


o .  29  596 
0.29  565 

0.29534 

0.29  502 

0.29471 


o .  29  440 
o .  29  408 

0.29377 
0.29  346 
0.29315 


0.29  283 


L.  Cotg.  led. I L.  Tang. 

68° 


9  95  366 
9  95360 
9  95  354 
9  95348 
9  95341 


9  95335 
9  95  329 
9  95323 
9  95317 
9  95  310 


9  95304 
9.95298 
9.95292 
9  95  286 
9  95  279 


9  95273 
9.95  267 
9  95  261 
9  95254 
9  95  248 


9  95242 
9  95236 
9  95229 
9  95223 
9,95217 


9  95  211 
9.95  204 
9.95  198 
9  95 192 
9  95  185 


9  95  179 
9  95  173 
9  95 167 
9  95 160 
9  95 154 


9.95 148 
9  95 141 
9  95 135 
9.95 129 
9  95 122 


9  95  "6 
9  95  "o 
9  95  103 
9,95097 
9.95090 


9.95084 
9.95078 
9.95071 
9.95065 
9  95059 


9.95052 
9.95046 
9  95039 
9  95033 
9.95027 


9.95  020 
9.95014 
9.95007 
9  95001 
9  94  995 


9,94988 


L.  Sin, 


d. 


60 

59 
58 
57 

55 
54 
53 
52 
51 
50 
49 
48 
47 

45 
44 
43 
42 
41 
40 
39 
38 

36 


25 
24 
23 
22 
21 
20 

19 
18 

17 
16 


Prop.  Pts. 


3a 

3 

2 

6 

4 

9 

6 

12 

8 

16 

0 

19 

2 

22 

4 

% 

6 

8 

26 

.1 

2 

.2 

5 

•3 

7 

■4 

10 

.5 

13 

.6 

■7 

W 

.8 

20 

•9 

23 

3» 

3.1 

6,2 

9  3 
12  4 

21  7 

24  8 
27.9 


as 

25 

50 

7-5 

10.0 

12  5 
15  o 
17  5 
20.0 
22.5 


a4 
24 

8 

2 
6 
o 

4 
8 
2 
6 


7 

I 

07 

2 

14 

•3 

2    I 

•4 

28 

i 

3  5 
42 

I 

S  6 

9 

6.3 

6 
2 

8 

4 
o 
6 
2 
8 
54 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


53 


w)7o 


9_ 
10 
I 

12 

13 
H 

\l 
\l 

20 

21 

22 
23 
24 

26 
27 
28 
29 


31 

32 

33 
31 

36 
37 
38 
39L 
40 

41 

42 

43 
44 

46 

47 
48 

49 

60 

51 
52 
53 
ii 
55 
56 
57 
58 
_59 
60 


L.  Sin, 


65  705 
65  729 

65  754 
65  779 
65  804 


65828 

65853 
65878 
65  902 
65927 


65952 
65976 
66  001 
66  025 
66  050 


66075 
66  099 
66  124 
66  148 
66  173 


9.66  197 
9.66  221 
9  66  246 
9  66  270 
9 .  66  295 


66319 

66343 
66368 
66392 
66  416 


66  441 
66465 
66489 
66513 
66  537, 


66  562 
66586 
66  610 
66  634 
66658 


66682 
66  706 
66  731 

66755 
66  779 


66803 
66827 
66851 
66875 
66899 


66  922 
66  946 
66  970 
66  994 
67018 


67  042 
67  066 
67  090 
67  "3 
67137 
67  i6i 


L.  Cos. 


24 
25 

2S 

24 

25 

24 
25 
25 
24 
25 
24 
25 
24 
24 
25 
24 
25 

24 
24 
25 
24 
24 

25 

24 
24 
24 
24 

25 

24 
24 
24 
24 
24 
24 
25 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 


L.  Tang. 


c.  d, 


70717 
70748 
70779 
70  810 
70  841 


70873 
70904 

70935 
70  966 
70997 


71  028 
71  059 
71  090 
71  121 
71  153 


71  184 
71  215 
71  246 
71  277 
71  308 


71  339 
71  370 
71  401 

71  431 
71  462 


71  493 
71  524 
71  555 
71586 
71  617 


71  648 
71  679 
71  709 
71  740 
71  771 


71  802 
71833 
71863 
71  894 
71  925 


72  017 
72  048 
72  078 


72  109 
72  140 
72  170 
72  201 
72231 


72  262 
72293 
72323 
72354 
72384 


72415 
72445 
72476 
72  506 
72537 


72567 


Cotg. 


3» 
31 
31 
31 

32 

31 
31 
31 
31 
31 
31 
31 
31 
32 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
31 
31 
31 
31 

31 
30 
31 
31 
31 
31 
30 
31 
31 
30 

31 
31 
31 
30 

31 

31 
30 
31 
30 
31 
31 
30 
31 
30 
31 
3? 
31 
30 
31 
30 


L.  Cotg. 


0.29  283 
0.29  252 
0.29  221 
0.29  790 
0.29  159 


0.29  127 
o .  29  096 
o . 29  065 

0.29034 

o .  29  003 


0.28  972 
0.28  941 
0.28  910 

0.28879 
0.28847 


0.28816 
0.28  785 
0.28754 

0.28  723 
o .  28  692 


0.28661 

o .  28  630 
o .  28  599 
0.28  569 
0.28538 


0.28  507 
o  28  476 
0.28  445 
0.28  414 

0.28383 


0.28  352 
0.28  321 
0.28  291 
0.28  260 
0.28  229 


0.28  198 
0.28  167 
0.28  137 
0.28  106 
0.28075 


o .  28  045 
0.28  014 

0.27983 
0.27  952 
0.27  922 


o  27  891 
0.27  860 
0.27  830 
0.27  799 
0.27  769 


c.d, 


0.27  738 
o  27  707 
o  27  677 
o  27  646 
0.27  616 


0.27585 
0.27555 

0.27  524 

0.27494 
0.27463 


0.27433 


L.  Tang. 

62^ 


L.  Cos. 


94891 
94885 
94878 
94871 
94865 


94988 
94982 
94  975 
94969 
94962 


94956 
94  949 
94  943 
94936 
94930 


94923 
94917 
94  91 1 
94904 
94898 


94858 
94852 
94845 
94839 
94832 


94  826 
94819 
94813 
94  806 

94  799 


94  793 
94786 
94780 
94  773 
94767 


94  760 
94  753 
94  747 
94740 
94  734 


94727 
94720 

94714 
94707 
94700 


94694 
94687 
94  680 

94674 
94667 


94  660 
94654 
94647 
94  640 

94634 


94627 
94  620 
94614 
94607 
94  600 


9-94  593 


L.  Sin. 


d. 


60 

59 
58 

57 

55 
54 
53 
52 
11 
60 

It 

47 
J^ 
45 
44 
43 
42 
41 
40 

39 
38 

_3i 
35 
34 
33 
32 
31 


Prop.  Pts. 


2M 

I 

2 

3 
4 

32 

6.4 

9.6 

12.8 

16  0 

6 

19   2 

8^ 

22.4 
25.6 

9 

28.8 

SI 

II 

9-3 

12.4 

155 
18.6 
21 .7 
24.8 
27.9 


30 

30 

6  o 

90 

12.0 

15  o 

18.0 

21.0 

24.0 

27.0 


as 

I 

2.5 

2 

50 

3 

7  5 

4 

10. 0 

•5 

12.5 

.6 

15  0 

.7 

17-5 

.8 

20.0 

9 

22.5 

H 

2.4 

4.8 

7.2 

9.6 

12.0 

14.4 

16.8 

19  2 

21.6 


6.9 
9.2 

ii? 

18.4 
20  7 


7 

I 

0.7 

2 

14 

3 

2.  I 

4 

2.8 

5 

3  5 

6 

4.2 

I 

56 

9 

6.3 

6 

0.6 
I  .2 
1.8 
24 

36 
4.2 

48 

5  4 


Prop.  Pfs. 


54 


TABLE  II 


28^ 


0      9  67  i6i 


I 
2 
3 

i 

7 
8 

9 

10 

II 
12 
13 

II 

;i 

17 

18 

19 
20 

21 
22 
23 
ii 

26 
27 
28 

80 

31 


L.  Sin. 


67185 
67208 
67  232 
67256 


67280 

67303 
67327 

67350 
67374 


9.67398 
9  67421 

9  67445 
9  67  468 

9  67  492 


9  67515 
9  67539 
0.67  562 
9.67586 
9  67  609 


9  67  633 
9  67  656 
9 .  67  680 
9.67  703 
9  67  726 


9  67  750 
9  67  773 
9.67  796 
9  67  820 
9  67  843 
9  67866 
^_      9  67  890 

32  967913 

33  967936 

34  9  67959 

36 


I  39 

40 

41 
42 

43 
44 

46 
47 
48 

49 
60 

51 

52 
53 
ii 

59 
60 


9  67  982 
9  68006 
9  68  029 
9  68  052 
9  68  075 
9.68098 
9  68  121 
9  68  144 
9  68  167 
9  68  190 


9  68  213 
9  68  237 
9 .  68  260 
9  68  283 
9  68  305 


9  68  328 
9  68351 
9  68  374 
9  68  397 
9  68  420 
"9  68443 
9.68466 
9.68489 
9.68  512 
9-68  534 


9  68  557 


L.  Cos. 


24 

23 

24 

24 

24 

23 

24 

23 

24 

24 

23 

24 

23 

24 

23 

24 

23 

24 

23 

24 

23 

24 

23 

23 

24 

23 

23 

24 

23 

23 

24 

23 

23 

23 

23 

24 

23 

23 

23 

23 

23 

23 

23 

23 

23 

24 

23 

23 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 


L.  Tang. 


72567 
72598 
72  628 
72659 
72689 


9  72  720 
9  72  750 
9  72  780 
9  72  811 
72  841 


c.d. 


72  872 
72  902 
72932 
72  963 
72993 


73023 

73 114 

73 144 


73 175 
73205 

73235 
73265 

73295 


73326 
73356 
73386 
73416 
73446 


73476 

73507 

73537 

9  73567 

9  12,  597 


9  73  627 
9  73  657 
9.73687 

73  717 
73  747 


73  777 
73807 
73837 
73867 
73897 


73927 
73  957 
73987 
74017 
74047 


9.74077 
9  74  107 
9  74  137 
9  74  166 

9  74  196 


74  226 
74256 
74286 
74316 
74  345 


9  74  375 


31 

30 

3» 
30 
31 
30 
30 
31 
30 
31 
30 
30 
31 
30 
30 
31 
30 
30 
30 
31 
30 
30 
30 
30 
31 
30 
30 
30 
30 
30 

31 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
29 
30 
30 
30 
30 
30 
29 
30 


L.  Cotg. 


0.27433 
0.27  402 
0.27  372 
o  27341 
0.27  311 


0.27  280 
0.27  250 
0.27  220 
0.27  189 
0.27  159 
0.27  128 
0.27  098 
0.27  068 
0.27  037 
0.27  007 


0.26  977 
o .  26  946 
0.26  916 

0.26886 
0.26856 


0.26  825 
0.26  795 
0.26  765 
0.26  735 
0.26  705 


0.26  674 
o .  26  644 
0.26  614 
0.26  584 
0.26554 


0.26  524 
o .  26  493 
o .  26  463 

0.26433 

o .  26  403 


L.  Cos. 


0.26373 
0.26343 
0.26  313 
o .  26  283 
0.26  253 


o .  26  223 

0.26  193 

0.26  163 
o  26  133 
0.26  103 


o  26  073 
o  26  043 
0.26  013 
o  25983 
o  25953 


25923 
25893 
25863 

25  834 

25  804 


0.25  774 
o  25  744 
0.25  714 
o  25  684 
0.25655 


0.25  625 


L.  Cotg.  c.  d.  L.  Tang 

61^ 


9  94  593 
9  94  587 
9  94  580 
9  94  573 
9  94567 


9  94  560 
9  94553 

9-94  546 
9  94540 
9  94  533 


9  94526 
9  94519 
9  94513 
9  94506 
9  94  499 


9  94492 
9  94485 
9  94  479 
9.94472 

9  94465 


9  94458 
9-94  451 
9  94  445 
9-94  438 
9  94431 


9  94424 
9  94417 
9.94410 
9.94404 
9  94  397 


d. 


9  94390 
9  94383 
9  94376 
9  94369 
9.94362 


9  94  355 
9  94  349 
9  94342 
9-94  335 
9.94328 


9  94321 
9  94314 
9  94307 
9.94300 

9  94293 


9 .  94  286 
9.94279 

9  94273 
9  94  266 

9  94259 


9  94252 

9  94245 
9  94238 
9  94  231 
9  94224 


9  94217 
9  94  210 
9  94  203 
9  94  196 
9  94  189 


9.94  182 


L.  Sin. 


60 

1 

55 
54 
53 
52 
SL 
60 
49 
48 
47 

45 
44 
43 
42 
_4i_ 
40' 
39 
38 

36 

35 
34 
33 
32 

30 

29 
28 
27 
26 


Prop.  Pts. 


31   1 

I 

.2 

C 

3 
4 

1 

9  3 
12.4 

•7 
.8 

9 

21.7 
24.8 
27.9 

30 
30 


6  O 

9  o 

12  C 
18  O 

21  .0 
24.0 
27.0 


29 
29 
58 
8.7 

II  6 
14  5 
17  4 
20.3 
23.2 
26.1 


34 

I 

2.4 

2 

4.8 

3 

7.2 

4 

9.6 

•  S 

12  0 

.6 

14  4 

.7 

16.8 

.8 

19.2 

9 

21.6 

23 

6.9 
92 
II. 5 

«3  8 
16  I 
18  4 
20.7 


7 

I 

0.7 

2 

14 

3 

2.1 

4 

2   8 

5 

3  5 

6 

42 

I 

S  6 

9 

6.3 

sa 

2.2 

4  4 
6.6 
8  8 
no 
13  2 

176 
19.8 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


55 


29°                                        1 

0 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

1.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

9  68557 

23 

9  74  375 

0 .  25  625 

9.94  182 

■fiO" 

I 

9 

68580 

9 

74405 

0.25595 

9 

94175 
94168 

S9 

2 

9 

68603 

9 

74  435 

0.25565 

9 

S8 

30 

3 

9 

68625 

23 
23 

9 

74465 

0.25535 

9 

94  161 

S7 

_4 

5 

9 

68648 

9 

74  494 

30 

0  25  506 

9 

94  154 

56 

SS 

2 

7 

io 
9  0 
12.0 

9 

68671 

9 

74524 

0.25476 

9 

94147 

6 

9 

68694 

9 

74  554 

0 .  25  446 

9 

94140 

S4 

4 

7 

9 

68716 

9  74  583 

29 

0.25417 

9 

94133 

S3 

5 

15  0 
18.0 

8 

9 

68739 

23 
22 

9  74613 

30 

0.25387 

9 

94126 

S2 

6 

9 
10 

9 

68  762 

9  74  643 

30 

0.25357 

9 

94  119 

51 
50 

I 

21.0 
24.0 

9  68  784 

9 

74673 

0.25327 

9 

94  112 

II 

9  68  807 

9 

74702 

29 

0.25  298 

9 

94  105        !    1 

49 

9 

27.0 

12 

9  68  829 

9 

74732 

0.25  268 

9 

94098 

48 

1 

13 

9  68  852 

23 
22 

9 

74762 

30 

0.25  238 

9 

94090 

47 

14 
IS 

9  68  875 

9 

74791 

30 

0 .  25  209 

9 

94083 

46 
4S 

1 

29 
2.9 

9  68  897 

9 

74821 

0.25  179 

9 

94076 

I 

i6 

9  68  920 

9 

74851 

30 

0.25  149 

9 

94069 

44 

2 

17 

9 

68942 

9 

74880 

29 

0.25  120 

9 

94062 

43 

3 

8-7 
II. 6 

i8 

9 

68965 

9 

74910 

30 

0.25  090 

9 

94055 

42 

4 

19 

2r 

9 

68987 

23 

9 

74  939 

29 
30 

0.25  061 

9 

94048 

41 
40 

•s^ 

145 
17  4 
20.3 

'41 

9 

69  010 

9 

74969 

0.25  031 

9 

94041 

21 

9 

69032 

9 

74998 

0.25  002 

9 

94034 

7 

39 

22 

9 

69055 

9 

75028 

0.24972 

9 

94027 

38 

23 

9 

69077 

9 

75058 

30 

0.24942 

9 

94020 

37 

24 

25 

9 

69  100 

22 

9 

75087 

29 

30 

0.24913 

9 

94012 

36 
3S 

1 

9 

69  122 

9 

75  117 

0.24883 

9 

94005 

23 

26 

9 

69  144 

9 

75  146 

29 

0.24  854 

9 

93998 

34 

T 

1:1 
69 
9.2 

27 

9 

69  167 

9 

75  176 

0 .  24  824 

9 

93991 

33 

0 

28 

9 

69  189 

9 

75  205 

29 

0.24795 

9 

93984 

32 

•3 
•  4 

29 

30 

9 

69  212 

22 

9 

75235 

30 
29 

0.24765 

9 

93  977 

31 
30 

9 

69234 

9 

75264 

0.24736 

9 

93970 

31 

9  69256 

9 

75294 

30 

0.24  706 

9 

93963 

29 

6 

32 

9  69279 

23 

9 

75323 

29 

0.24677 

9 

93  955 

28 

y 

16. 1 

33 

9 

69301 

9 

75  353 

30 

0.24647 

9 

93948 

27 

g 

18.4 

34 

3S 

9 

69323 

22 

9 

75382 

29 
29 

0.24  618 

9 

93  941 

26 
25 

.9 

20.7 

9 

69345 
69  368 

9 

75  411 

0.24589 

9 

93  934 

1 

36 

9 

9 

75441 

30 

0.24559 

9 

93927 

24 

1 

37 

9 

69390 

9 

75470 

29 

0.24530 

9 

93920 

23 

aa 

3« 

9 

69412 

9 

75  500 

30 

0 .  24  500 

9 

93912 

22 

.  I 

2.2 

39 
40 

9 

9 

69434 

22 

9 

75529 

29 
29 

0.24471 

9 

93905 

21 
20 

.2 

.3 

44 
6.6 

69456 

9 

•75558 

0.24442 

9 

93898 

41 

9 

69479 

23 

9 

75588 

30 

0.24412 

9 

93?9i 

19 

•  4 

8.8 

42 

9 

69501 

9 

75617 

29 

0.24383 

9 

93884 

18 

II. 0 

43 

9 

•69  523 

9 

75647 

30 

0.24353 

9 

93876 

17 

6 

13.2 

44 
45 

9 

69545 

22 

9 

•  75  676 

29 
29 

0.24324 

9 

93869 

16 
15 

:l 

17.6 

9 

69567 

9 

•75  705 

0.24295 

9 

93  862 

46 

9 

.69589 

9 

•  75  735 

30 

0.24  265 

9 

93  855 

14 

9 

19.8 

47 

9 

69  611 

9 

■  75  764 

29 

0.24236 

9 

93847 

13 

1 

48 

9 

69633 

9 

•75  793 

29 

0 .  24  207 

9 

93840 

12 

1 

49 
50 

_9 
9 

69655 

22 

9 

.75822 

29 
30 

0.24  178 

9 

93833 

II 
10 

I 

8 

0.8 

7 

0.7 

.69677 

9 

.75852 

0.24  148 

9 

.93826 

SI 

9 

.69699 

9 

.75881 

29 

0.24  119 

9 

93819 

9 

2 

lb 

14 

S2 

9 

.69  721 

9 

75910 

29 

0 .  24  090 

9 

93  811 

8 

3 

2.4 

2.1 

S3 

9 

69  743 

9 

75  939 

29 

0.24061 

9 

93804 

7 

4 

32 

2.8 

54 
SS 

9 

69  765 

22 

9 

75  969 

30 
29 

0 .  24  03 1 

9 

93  797 

b 
5 

I 

It 

73 

3  b 
4.2 

49 

9 

69787 

9 

75998 

0 .  24  002 

9 

93789 

S6 

9 

69809 

9 

.76027 

29 

0.23973 

9 

93  782 

4 

57 

9 

69831 

9 

76  056 

29 

0.23944 

9 

W 

3 

SB 

9 

69853 

9 

.  76  086 

30 

0.23914 

9 

8 
7 

2 

y  1 

59 

9 

.69875 

22 

9 

.76115 

29 
29 

0.23885 

9 

93760 

I 
0 

9 

.69897 

9 

76144 

0.23856 

9 

93  753 

L.  Cos. 

~ 

L.  Cotg. 

c.d. 

L.  Tangi 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

60^                                       1 

56 


TABLE  II 


30^ 


0 

I 

2 

3 

i 

7 
8 

_9_ 

10 

II 

12 

13 

M 


19 
20 

21 
22 
23 
24 

26 
27 
28 
29 

30 

31 
32 
33 
34 

36 
37 
3« 
39 
40 
41 
42 
43 
44 

46 

47 
48 

49 

60 

51 

52 
53 

il 

i2. 
00 


L.  Sin, 


69897 
69919 
69941 
69963 
69  984 


70  006 
70028 
70  050 
70  072 
70093 


70  115 
70137 

70159 
70  180 
70  202 


70  224 

70245 
70  267 
76288 
70310 


70332 
70353 
70375 
70396 
70418 


70439 
70461 
70  482 
70504 
70525 


70547 
70568 
70590 
70  611 
70633 


70654 
70675 
70697 
70  718 
70  739 


70  761 
70  782 
70803 
70  824 
70  846 


70867 
70888 
70  909 
70931 
70952 


70973 
70994 
71  015 
71  036 
71058 


71  079 
71  100 
71  121 
71  142 
71  163 


71  184 


L.  Cos. 


d. 


L.  Tally. 


76  144 
76  173 
76  202 
76231 
76  261 


76  290 
76319 
76348 
76377 
76  406 


76435 
76464 

76493 
76  522 

76551 


76580 
76  609 
76639 
76668 
76697 


76725 
76  754 
76  1^2> 
76812 
76841 


76870 
76899 
76928 

76957 
76986 


77015 

77044 
77073 
77  101 
77  130 


77  159 
77188 
77217 
77246 
77274 


77303 
77332 
77361 
7739c 
77418 


77  447 
77476 
77505 
77  533 
77562 


77591 
77619 
77648 

77677 
77706 


77  734 
77  763 
77791 
77  820 
77^^49 


77877 


d.     L.  Cotg. 


c.  d. 


L.  Cotg. 


o  23  856 
o  23  827 
0.23  798 
0.23  769 
0.23  739 


0.23  710 
0.23  681 
0.23  652 
0.23  623 
0.23  594 


0.23565 
0.23  536 
0.23  507 
0.23478 
0.23449 


o . 23  420 
0.23391 
0.23  361 
0.23332 
0.23303 


0.23  275 
0.23  246 
0.23  217 
0.23  188 
o  23  159 


0.23  130 
0.23  lOI 

0.23  072 

0.23043 
0.23014 


0 .  22  985 
O  22  956 
O  22  927 
O  22  899 
0.22  870 


c.  d. 


O  22  841 
O  22  812 
O  22  783 
0.22  754 
0.22  726 


0.22  697 
0 .  22  668 
0 .  22  639 
0.22  610 
O  22  582 


0.22553 
0.22  524 
0 .  22  495 
O  22  467 
0 .  22  438 


0.22  409 
O  22  381 
O  22  352 
0 .  22  323 
O  22  294 


0.22  266 
0.22  237 
0 .  22  209 
O  22  180 
0.22  151 


0.22  123 


L.  Tang. 

59^ 


L.  Cos. 

d. 

60 

9  93  753 

9 

93  746 

59 

9 

93738 

58 

9 

93  731 

S7 

9 

93  724 

7 

56 

9 

93  717 

55 

9 

93  709 

54 

9 

93  702 

S^ 

9 

93695 

52 

9 

93687 

51 
50 

9  93  680 

9  93  673 

49 

9  93665 

48 

9 

93  658 

47 

9 

93650 

46 
4S 

9 

93643 

9 

93636 

44 

9 

93  628 

4^ 

9 

93621 

42 

9 

93614 

41 
40 

9 

93606 

9 

93  599 

39 

9 

93591 

S8 

9 

93584 

37 

9 

93  577 

36 

35 

9 

93569 

9 

93562 

34 

9 

93  554 

33 

9 

93  547 

32 

9 

93  539 

31 
30 

9 

93532 

9 

93525 

29 

9 

93517 

28 

9 

93510 

27 

9 

93502 

26 

25 

9 

93  495 

9 

93487 

24 

9 

93480 

23 

9 

93472 

22 

9 

93465 

21 
20 

9 

93  457 

9 

93450 

19 

9 

93442 

18 

9 

93  435 

17 

9 

93427 

16 
15 

9 

93420 

9 

93412 

14 

9 

93405 

13 

9 

93  397 

12 

9 

93390 

II 
10 

9 

93382 

9 

93  375 

9 

9 

93367 

8 

9 

93360 

7 

9 

93352 

6 

5 

9 

93  344 

9 

9Z2,i1 

4 

9 

93329 

3 

9 

93322 

d. 

2 

9 

93314 

0 

9 

93307 

L.  Sin. 

Prop.  Pts. 


30 

I 

3  0 

2 

6.0 

3 

9.0 

4 

12.0 

15.0 

6 

18  0 

7 

21.0 

8 

24.0 

9 

27.0 

2.9 
8 

7 
6 

5 
4 


5 

8 
II 
14 
17 
20.3 
23.2 
26.1 


28 
28 


8 

.1 

0.8 

.2 

r  6 

3 

2.4 

4 

32 

1; 

4.0 

6 

48 

i 

l\ 

9 

7-2! 

23 
2    2 

4  4 
6  6 
8.8 

II. o 

13.2 

17  6 
19.8 


ai 
2.1 

42 

8.4 

12.6 

14  7 
16  8 
18.9 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


57 


31°                                       1 

L.  Sin. 

d. 

L.  Tang.  c.  d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

9  71  184 

977877     ,„ 

0.22  123 

9  93307 

g 

60 

I 

9  71  205 

9  77  906 

29 

28 

0 .  22  094 

9.93299 

3 

59 

2 

9  71  226 

9  77  935 

0 .  22  065 

9  93291 

58 

33 

^ 

9  71  247 

9  77  963 

0.22037 

9  93  284 

Q 

57 

T 

2  9 

9  71  268 

21 

9  77  992 

28 
29 
28 

0  22  008 

9  93276 

7 
8 

56 

55 

2 

3 

9  71  289 

9  78  020 

0,21  980 

9.93269 

6 

9  71  310 

21 

9  78049 

0.21  951 

9.93261 

8 

54 

4 

II  6 

7 

9  71  331 

9  78  077 

0  21  923 

9-93  253 

53 

14  5 

8 

9 

10 

9  71  352 
9  71  373 

21 
20 

9  78  106 
9  78135 

29 
28 

0.21  894 
0.21  865 

9.93246 
9-93  238 

8 
8 

52 
51 
60 

7 
8 

17  4 
20  3 
23.2 

9  71  393 

9  78  163 

0  21  837 

9  93230 

II 

9  71  414 

9.78  192 

28 

0  21  808 

9.93223 

8 

49 

9 

26.1 

12 

9  71  435 

9  78  220 

0  31  780 

9  93215 

8 

.48 

1 

n 

9  71  456 

9  78  249 

0  21  751 

9.93207 

47 

15 

9  71  477 

21 

9.78277 

29 
28 

0  21  723 

9  93200 

8 
8 

46 

45 

28 
2  8 
^.6 
8.4 

9  71  498 

9.78306 

0.21  694 

9  93  192 

I 

16 

9 

71  519 

9  78  334 

0.21  666 

9  93  184 

44 

2 

•7 

9 

71  539 

9  78363 

29 
28 
28 
29 
28 

0  21  637 

9  93  177 

8 
8 
7 
8 
8 

43 

3 

18 

9 

71  560 

^ 

9.78391 

0.21  609 

9  93  169 

42 

4 

II  .2 

19 
20 

9 

71  581 

21 

9.78419 

0.21  581 

9  93  161 

41 
40 

I 

14.0 
16.8 
19  6 
22  4 
25.2 

9 

71  602 

9.78448 

0.21  552 

9  93  154 

21 

9 

71  622 

9.78476 

0.21  524 

9  93  146 

39 

22 

9 

71  643 

9  78505 

29 
28 

0.21  495 

9  93  138 

38 

23 

9 

71664 

9  78533 

0.21  467 

9  93  131 

7 
8 
8 

37 

•y 

24 

2S 

9 

71685 

20 

9.78562 

29 
28 
28 

0.21  438 

9  93  123 

36 

35 

1 

9 

71  705 

9  78  590 

0.21  410 

9  93  "5 

26 

9 

71  726 

9.78618 

0.21  382 

9.93  108 

7 

34 

2  I 

27 

9 

71  747 

9  78  647 

29 
28 

0.21  353 

9.93  100 

8 
8 
7 
8 
8 
8 

33 

4.2 

8.4 

10  5 

28 

9 

71  767 

9.78675 

0.21  325 

9.93092 

32 

3 
•  4 

.5 

29 

30 

9 

71788 

21 

9  78  704 

29 
28 
28 

0.21  296 

9.93084 

31 
30 

9 

71  809 

9.78732 

0.21  268 

9.93077 

31 

9 

71  829 

9,78  760 

0.21  240 

9.93069 

29 

.6 

12.6 

32 

9  71  850 

9,78789 

29 
28 
28 
29 
28 
28 

0.21  211 

9.93061 

28 

.7 

14  7 

,1S 

9  71  870 

9.78817 

0.21  183 

9-93  053 

27 

.8 

16.8 

34 

3S 

9  71  891 

20 

9.78845 

0.21  155 

9.93046 

8 

26 

25 

9 

18.9 

9  71  911 

9.78874 

0.21  126 

9-93  038 

1 

^^ 

9  71  932 

9.78902 

0.21  098 

9.93030 

24 

1 

37 

9  71  952 

9.78930 

0.21  070 

9.93022 

8 

23 

•   20 

3« 

9 

71  973 

9  78959 

29 
28 
28 
28 

0.21  041 

9.93014 

22 

J 

2.0 

39 
40 

9 

71  994 

20 

9  78987 

0.21  013 

9.93007 

8 

Q 

21 
20 

.2 
■  3 

40 
6.0 

9 

72014 

9.79015 

0 .  20  985 

9.92999 

41 

9 

72034 

9  79043 

0.20957 

9.92991 

Q 

19 

•  4 

8.0 

42 

9 

72055 

9 • 79  072 

29 

0.20  928 

9.92983 

18 

.5 

10.0 

43 

9 

72075 

9.79  100 

0 .  20  900 

9.92976 

8 
8 
8 

Q 

17 

.6 

12.0 

_4£_ 
4S 

9 

72096 

20 

9.79  128 

28 

0.20  872 

9.92  968 

16 
15 

•7 
.8 

14.0 
16.0 
18.0 

9 

72  116 

9  79156 
9  79  185 

0 .  20  844 

9.92960 

46 

9 

72  137 

29 
28 

0.20  815 

9.92952 

14 

•9 

47 

9 

72  157 

9.79213 

0.20  787 

9.92944 

D 

13 

1 

48 

9 

72  177 

9  79  241 

0.20759 

9.92936 

12 

1 

49 
50 

9 

72  198 

20 

9  79  269 

28 
28 

0  20  731 

9.92929 

7 

8 
8 
8 
8 
8 
8 

10 

.1 

8 
0.8 

7 
07 

9 

.72218 

9.79297 

0.20  703 

9.92-921 

51 

9 

.72238 

9  79  326 

29 

0 . 20  674 

9  92913 

9 

2 

I.b 

14 

•ia 

9 

72  259 

9  79  354 

0 .  20  646 

9.92905 

8 

3 

2.4 

2.  I 

S3 

9 

.72  279 

9.79382 

28 

0.20618 

9.92897 

7 

4 

32 

2   8 

54 
SS 

9 

72  299 

21 

9.79410 

28 
28 

0 .  20  590 

9  92  889 

6 

5 

6 

4.0 

3  b 

4  2 
4  9 

9  72  320 

9 -79  438 

0.20  562 

9.92881 

,S6 

9  72  340 

9.79466 

28 

0.20534 

9  92874 

7 

4 

I 

S7 

9.72360 

9  79  495 

29 

0 .  20  505 

9.92866 

3 

Sb 

9  72  381 

9  79523 

28 

0.20477 

9.92858 

2 

y  1 

/  ■^  *'  Ji 

59 
00 

9  72  401 

20 

9  79551 

28 
28 

0 .  20  449 

9  92  850 

8 

I 

0 

9.72421 

9  79  579 

0.20421 

9 .  92  842 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

Prop.  Pts. 

58^ 

1 

58 


TABLE  II 


32°                                       1 

— 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

60 

Prop.  Pte. 

9.72421 

9-79  579 

28 

0.20  421 

9.92842 

8 
8 
8 
8 
7 
8 

9.72441 

9.79607 

0.20393 

9.92834 

59 

2 

9  72  461 

9  79635 

28 

0.20  365 

9.92  826 

58 

29      38 
2.9    2.8 
58    5-6 
8.7    8.4 

3 
4 

s 

9  72482 
9  72  502 

20 
20 

9.79663 
9.79691 

28 
28 
28 

0  20337 

0  20  309 

9.92  818 
9  92  810 

u 

55 

2 
3 

9.72522 

9.79719 

0.20  281 

9  92  803 

6 

9.72542 

9-79  747 

0.20  253 

9  92  795 

8 

54 

4 

II .6   II  .2 

7 

9  72  562 

9  79  776 

28 

0.20  224 

9  92787 

53 

14. c 

I4.0 

8 

9  72582 

9.79804 

28 

0.20  196 

9  92  779 

52 

6 

17  4   16  8 

9 
10 

9 .  72  602 

20 

9.79832 

28 

0.20  168 

9.92771 

8 
8 
8 
8 
8 
8 
8 
8 
8 

51 
50 

•7 
8 

20.3    19.6 
23  2  22.4 

9  72  622 

9 .  79  860 

0.20  140 

9  92763 

II 

9.72643 

9.79888 

28 

0.20  112 

9  92  755 

49 

9 

26.] 

I252 

12 

9  72  663 

9.79916 

28 

0.20084 

9  92  747 

48 

1 

M 

9  72683 

9-79  944 

28 

0.20  056 

9  92  739 

47 

1 

14 

9  72  703 

20 

9.79972 

28 
28 

0 .  20  028 

9.92  731 

46 

45 

I 

37 

2.7 

9  72  723 

9.80000 

0.20000 

9.92  723 

i6 

9  72  743 

9.80028 

28 

0.19972 

9.92  715 

44 

.2 

5  4 

17 

9.72  763 

9 .  80  056 

0   19944 

9.92  707 

43 

3 

8.1 

i8 

9  72  783 

9.80084 

28 
28 
28 

0.19  916 

9.92699 

42 

4    10. 0 

'9 
20 

9.72803 

20 

9.80  112 

0.19888 

9.92691 

8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 

41 
40 

I  lit 

9.72823 

9.80  140 

0.19  860 

9.92683 

21 

9.72843 

9.80  168 

0.19  832 

9  92675 

39 

22 

9  72863 

20 

9.80195 

0.  19  805 

9,92667 

38 

9    24.3 

2^ 

9  72883 

9.80223 

28 

0   19777 

9  92659 

37 

25 

9.72902 

20 

9.80251 

28 
,0 

0.19749 

9  92651 

36 

35 

9.72922 

9.80279 

0.19  721 

9.92643 

21   1     90    1 

26 

9.72942 

9.80307 

_Q 

0   19  693 

9  92635 

34 

J 

0 

I      20 

27 

9.72962 

9-80335 

28 
28 
28 

0,19  665 

9.92627 

33 

0 

4 

6. 

8. 

10. 

2  40 

3  6.0 

4  8.0 

5  10.0 

28 
29 

30 

9.72982 
9.73002 

20 
20 

9.80363 
9.80391 

0.19637 
0.19  609 

9  92  619 
9.92  611 

32 
31 
30 

3 
4 

9.73022 

9.80419 

0.19  581 

9  92  603 

^i 

9.73041 

9.80447 

o- 19  553 

9  92  595 

29- 

5 

12. 

6   12.0 

32 

9.73061 

9.80474 

27 
28 

0.19526 

9.92587 

28 

.7 

14 

7   14  0 

33 

9.73081 

9.80  502 

0.19498 

9  92579 

8 
8 
8 

27 

.8 

16. 

8   16.0 

34 

35 

9  73  loi 

20 

9.80530 

28 

28 

0.19470 

9.92571 

26 

25 

9 

18. 

9  18.0 

9  73  121 

9.80558 

0.19442 

9  92563 

36 

9  73  140 

9.80586 

28 
28 
27 

28 
28 

„Q 

0.19  414 

9  92555 

24 

1 

37 

9.73  160 

9.80  614 

0.19386 

9.92546 

9 
8 
8 
8 
8 
8 
8 
8 
8 

23 

19 

1      9 

3« 
39 
40 

41 

9.73180 
9.73200 

20 
20 

9 .  80  642 
9.80669 

o.  19  358 
0.19  331 

9  92538 
9  92530 

22 
21 
20 

19 

.1 

.2 

3 
.4 

3 

5- 
7- 

9     0.9 

8     1.8 
7     2.7 
6     3.6 

9.73219 
9 -73  239 

9.80697 
9-80725 

0.19303 
0.19275 

9.92522 
9.92514 

42 

9  73259 

9-80753 
9.80  781 

28 

0.19247 

9.92506 

18 

.5 

9 

5     4  5 

43 

9.73278 

0.19  219 

9.92498 

17 

.6 

II 

4     5  4 

44 
4S 

9.73298 

20 

9.80808 

28 
28 
28 

0.19  192 

9.92490 

16 
15 

■I 

13 
15 

3     6.3 
2     7.2 

9  73318 

9.80836 

0.19  164 

9  92  482 

46 

9  73  337 

^9 

9.80864 

0.19  136 

9  92473 

9 
8 
8 
8 
8 
8 
8 

14 

9 

17 

I     8.1 

47 

9  73  357 

9.80892 

0.19  108 

9.92465 

13 

1 

48 

9-73  377 

9.80919 

27 
28 
28 

28 

0.19  081 

9  92457 

12 

1 

49 
50 

9  73396 

^9 
20 

9.80947 

0.19053 

9.92449 

II 
10 

I 

8 
0. 

7 

8     0.7 

9  73416 

9.80975 

0.19025 

9.92441 

51 

9  73  435 

19 

9.81  003 

0.18997 

9  92433 

9 

.2 

I. 

6     1.4 

52 

9  73  455 

9.81  030 

27 

28 
28 
27 
28 
28 

0. 18  970 

9.92425 

8 

3 

2. 

4     21 

S3 

9  73  474 

^9 

9.81  058 
9.81  086 

0.18942 

9.92416 

9 
8 
8 
8 
8 

7 

-4 

3 

2     28 

54 

9  73  494 

19 

0.18914 

9 .  92  408 

6 

5 

i 

4 
4 

0     3  5 
8     42 
6     49 

\  is 

9  73513 

9.81  113 

0.18887 

9.92400 

56 

9  73  533 

9  81  141 

0.18859 

9.92392 

4 

■'J 
.8 

6: 

^S^ 

9  73552 

*9 

9.81  169 

0.18  831 

9  92384 

3 

9-73  572 

9.81  196 

27 

28 
28 

0.18804 

9  92376 

2 

y 

y 

59 
60_ 

9  73591 

19 
20 

9  81  224 

0.18  776 

9.92367 

9 
8 

I 
_0^ 

9.73  611 

9.81  252 

0.18  748 

9  92  359 

L.  Cos. 

d. 

L.  Cotg.  Ic.  d. 

L.  Tang. 

L.  Sin. 

d. 

Prop.  Pte. 

57°                                        1 

LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


59 


33° 

1 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

"60" 

Prop.  Pt8. 

9-73611 

19 

9  81  252 

27 

0.18  748 

9  92359 

g 

I 

9  73630 

20 

9  81  279 

28 

0.18  721 

9  92351 

8 

59 

2 

9-73  650 

IQ 

9  81  307 

28 

0.18  693 

9  92343 

0 

58 

38 

87 

2.7 

5  4 

3 

9.73669 

ao 

981335 

27 
28 
28 

0.18665 

9  92335 

57 

T 

2.8 
5.6 
8.4 

4 

9.73689 

19 
19 

9  81  362 

0.18638 

9  92326 

8 

8 
8 

56 

55 

.2 

.3 

9  73  708 

9.81390 

0  18  610 

9  92318 

6 

9  73727 

20 

9  81  418 

0.18  582 

9,92310 

54 

4 

II. 2 

10.8 

7 

9  73  747 

19 

9.81445 

28 

0.18555 

9,92302 

9 

8 

53 

14.0 

13  5 

8 

9  73  766 

9  81473 

0.18527 

9.92  293 

52 

.6 

16.8 

16.2 

9 

9  73785 

20 

9  81  500 

28 

0.18  500 

9  92285 

8 

51 

•7 

19.6 

18  9 

10 

9  73805 

10 

9.81  528 

28 

0.18472 

9.92277 

8 

50 

.8 

22.4 

21.6 

II 

9  73  824 

9  81  556 

27 
28 

0.18444 

9.92269 

9 

8 

49 

9 

25.2 

243 

12 

9  73843 

20 

9  81  583 

0.18  417 

9.92  260 

48 

1 

n 

9  73863 

19 
19 

9  81  611 

0.18389 

9.92252 

47 

15 

9  73  882 

9.81  638 

28 

0.18362 

9  92244 

9 

8 

46 
45 

1 

30 
2.0 

9  73901 

9.81  666 

0.18334 

9  92235 

I 

.16 

9.73921 

19 

9  81  693 

28 

0.18  307 

9  92227 

8 

44 

8.0 

17 

9  73940 

9  81  721 

0.18  279 

9.92219 

8 

43 

•3 

18 

9  73  959 

19 

9.81  748 

27 
28 
27 
,0 

0.18  252 

9  92  211 

42 

•4 

19 
20 

9  73978 

^9 
19 
20 

9.81  776 

0. 18  224 

9  92  202 

9 

8 
g 

41 
40 

'\ 

10.0 
12  0 
14  0 
16  0 

9  73  997 

9  81  803 

0.18  197 

9.92  194 

21 

9  74017 

19 
19 
19 
19 

9  81  831 

0.18  169 

9.92  186 

39 

22 

9.74056 

9  81  858 

28 

0.18  142 

9.92  177 

8 

38 

n 

18  0 

23 

9  74055 

9.81  886 

0.18  114 

9.92  169 

0 

37 

24 
25 

9  74  074 

9  81  913 

28 

0.18087 

9.92  161 

9 

8 

_36_ 
35 

1 

9  74093 

9  81  941 

0. 18  059 

9.92152 

19 

26 

9  74  "3 

19 
19 
19 
19 

9  81  968 

28 

0. 18  032 

9.92  144 

8 

34 

¥ 

3:1 

n 

27 
28 
29 
30 

9  74  132 
9  74  151 
9.74170 

9  81  996 
9  82  023 
9  82  051 

27 
28 
27 
28 

0.18004 
0.17977 
0.17949 

9.92  136 
9.92  127 
9.92  119 

9 
8 
8 

33 
32 
31 
30 

.2 

•3 
•4 

•  S 

9  74  189 

9.82078 

0.17922 

9.92  III 

31 

9.74208 

19 

9.82  106 

0.17894 

9.92  102 

0 

29 

.6 

I 

r  1 

32 

9.74227 

19 

9  82133 

28 

0.17867 

9.92094 

8 

28 

.7 

i^.\     1 

33 

9.74246 

^9 

9.82  161 

0.17839 

9.92086 

27 

.8 

)  2 

34 
35 

9.74265 

19 
»9 

9  82  188 

27 
27 
28 

0.17  812 

9.92077 

9 
8 

26 

25 

9 

I 

I 

9,74284 

9  82  215 

0.17785 

9.92069 

1 

36 

9  74303 

19 

9,82243 

0.17757 

9.92060 

8 

24 

1 

37 

9.74322 

9  82  270 

28 

0.17730 

9.92052 

8 

23 

18 

3« 

9  74341 

9.82  298 

0.17  702 

9.92044 

22 

1 

I  8 

39 
40 

9  74  360 

19 

9  82325 

27 
27 

28 

0,17675 

9  92035 

8 

21 
20 

2 
.3 

36 

5  4 

9  74  379 

9.82352 

0.17648 

9.92027 

41 

9  74398 

9  82  380 

0.17  620 

9.92  018 

19 

.4 

7.2 

42 

9  74417 

^9 

9  82  407 

27 
28 

0.17593 

9.92  010 

18 

9.0 

43 

9  74436 

^9 

9.82435 

0.17565 
0.17538 

9  92  002 

17 

6 

10.8 

44 

4S 

9  74  455 

»9 
»9 

9 .  82  462 

27 
27 
28 

9  91  993 

9 
8 

lb 
15 

i 

12.6 
14.4 

9  74  474 

9.82489 

0.175" 

9.91  985 

46 

9  74  493 

19 

9.82517 

0,17483 

9,91976 

9 
8 

14 

•9 

16.2 

47 

9  74512 

»9 

9.82  544 

27 

0.17456 

9.91  968 

13 

1 

48 

9  74  53^ 

19 
18 

9  82571 

27 
28 
27 

0.17429 

9  91  959 

8 

12 

1 

49 
50 

9  74  549 

19 

9.82599 

0.17  401 

9  91  951 

9 
0 

II 
10 

.  I 

9 
0.9 

8 

0.8 

9 • 74  568 

9.82  626 

0.17374 

9  91  942 

51 

9  74  587 

19 

9.82653 

27 

0.17347 

9  9i  934 

9 

.2 

1.8 

1.6 

52 

9.74606 

19 

9.82681 

0.17319 

9.91  925 

8 

8 

•3 

2.7 

2.4 

S3 

9  74  625 

19 

9.82  708 

27 

0.17  292 

9.91  917 

7 

.4 

36 

32 

54 

55 

9.74644 

19 
t8 

9  82735 

27 
27 

0.17265 

9,91  908 

9 
8 

6 

5 

i 
I 

9 

4.5 
5  4 
6.3 
7.2 
8.1 

n 

72 

9.74662 

9.82  762 

0.17  238 

9.91  900 

56 

9  74681 

19 

9.82  790 

0. 17  210 

9.91  891 

8 

9 
8 

4 

57 
58 

9 . 74  700 
9  74  719 

19 
19 
18 

19 

9  82817 
9.82844 

27 
27 

0.17  183 
0  17  156 

9,91883 
9,91  874 

3 
2 

59 
60 

9  74  737 

9.82871 

27 

28 

0.17  129 

9,91  866 

9 

I 
0 

9  74756 

9.82899 

0.17  lOI 

9  91  857 

L.  Cos. 

T" 

L.  Cotg. 

™d. 

L.  Tang* 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

56° 

1 

60 


TABLE  II 


34°                                       1 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

60 

Prop.  Pts. 

"F 

9-74  756 

9.82  899 

0.17  lOI 

9  91  857 

8 

I 

9 

74  775 

9 

82  926 

0.17074 

9.91  849 

5Q 

2 

3 

9 
9 

74  794 
74812 

18 

9 
9 

82953 
82  980 

27 
28 

0.17047 
0.17  020 

9.91  840 
9.91  832 

9 
8 

58 

57 

28 

2.8 
5.6 
8.4 

27 

4 

9 

74831 

19 
18 

9 

83008 

27 

0.16  992 

9.91  823 

9 
8 

56 

55 

.2 

•3 
.4 

2.7 

9 

74850 

9 

83035 

0. 16  965 

9.91  815 

6 

9 

74868 

9 

83062 

0.16938 

9.91  806 

9 
8 

54 

II  2 

10  8 

7 

9 

74887 

9 

83089 

28 

0.16911 

9.91  798 

^^ 

14.0 

8 

9 

74906 

18 

9 

^Z  117 

27 
27 

0.16883 

9.91  789 

9 
8 

9 

52 

t6  8 

9 
10 

9 

74924 

19 

9 

83 144 

0.16856 

9.91  781 

51 

oO 

•7 
.8 

19.6 

22.4 

189 
21.6 

9 

74  943 

9 

83 171 

0.16  829 

9.91  772 

II 

9 

74961 

9- 

83198 

0.16  802 

9.91  763 

9 
8 

4Q 

.9 

25.2 

24  3 

12 

9 

74980 

9 

83225 

0.16775 

9  91  755 

48 

n 

9 

74  999 

18 

9 

l^  ^p 

28 

0. 16  748 

9  91  746 

9 
8 

9 

47 

14 
15 

9 

75017 

19 

t8 

9 

2,z  280 

27 

0.16  720 

9  91  738 

46 

45 

! 

26 
26 

9 

75036 

9 

83307 

0.16  693 

9  91  729 

.1 

16 

9 

75054 

9 

83334 

0. 16  666 

9.91  720 

9 
8 

44 

.2 

M 

17 

9 

75073 

18 

9 

83  361 

0.16  639 

9.91  712 

43 

•3 

18 

9 

75091 

9 

83  388 

0. 16  612 

9.91  703 

9 
8 

9 

42 

.4 

10.4 

19 
^0" 

9 

75  no 

18 

9 

83415 

27 

28 

0.16  585 

9.91  695 

41 
40 

i 

13.0 

;§2 

9 

75128 

9 

83442 

0. 16  558 

9.91  686 

21 

9 

75  147 

18 

9 

83470 

27 
27 
27 
27 

0.16  530 

9.91677 

9 
8 

30 

20  8 

22 

9 

75  165 

9 

83497 

0.16  503 

9.91  669 

38 

23  4 

23 

9 

75184 

^9 

9 

83524 

0.16  476 

9.91  660 

9 

37 

•y 

24 

2S 

9 

75202 

19 

9 

83  551 

0  16449 

9,91  651 

9 
8 

36 

3S 

1 

9 

75221 

9 

83  578 

0. 16  422 

9.91  643 

X9 

26 

9 

75239 

9 

83605 

0.16395 
0.16368 

9.91  634 

9 

34 

^ 

I  -        1 

27 

9 

75258 

19 

18 

9 

83632 

9.91  625 

9 
8 

33 

0 

\ 

28 

9 

75276 

18 
19 

t8 

9 

83659 
83686 

0.16  341 

9.91  617 

32 

3 

•4 

>.7 
r6 

)  5 

29 

30 

9 

75294 

9 

27 

0.16  314 

9.91  608 

9 
9 
8 

31 
30 

9 

75313 

9 

83  713 

0.16287 

9-91  599 

31 

9 

75331 

9 

83740 

28 

0. 16  260 

9  91  591 

29 

(y 

I 

.4 

32 

9 

75350 

18 
18 
19 

9 

83  768 

0.16232 

9  91  582 

9 

28 

7 

I' 

r'i 

33 

9 

75368 

9 

83  795 

0,16  205 

9  91  573 

9 

27 

8 

1 

52 

7.1 

34 
35 

9 

75386 

9 

83822 

27 

0.16  178 

9  .91  565 

9 

26 
25 

9 

9 

75405 

9 

83  849 

0.16  151 

9  91  556 

1 

Z^ 

9 

75423 

18 

9 

83876 

0.16  124 

9  91  547 

9 

24 

1 

37 

9 

75441 

,Q 

9 

83  903 

0.16  097 

9  91  538 

9 
8 

23 

18 

3« 

9 

75  459 

9 

83930 

0.16  070 

9  91  530 

22 

I  8 

39 
40 

9 

75478 

^9 
18 

9 

83957 

27 

0. 16  043 

9.91  521 

9 
9 
8 

21 
20 

.2 
.3 

36 
5  4 

9 

75496 

9 

83984 

0  16  016 

9.91  512 

41 

9 

75514 

9 

84  on 

0.15989 

9.91  504 

19 

.4 

72 

42 

9 

75  533 

9 

84038 

0.15  962 

9  91  495 

9 

18 

.«; 

9.0 

43 

9 

75551 

18 
18 
18 

9 

84065 

o- 15  935 

9.91  486 

9 

17 

.6 

10.8 

44 
4.S 

9 

75569 

9 

84092 

27 

0.15908 

9.91477 

9 
8 

16 

i 

12  6 
14  4 

9 

75587 

9 

84  119 

0.15  881 

9.91  469 

4b 

9 

75605 

9 

84  146 

27 

0.15854 

9.91  460 

9 

14 

9 

16.2 

47 

9 

75624 

19 
18 

9 

84173 

27 

0.15827 

9  91  451 

9 

13 

1 

48 

9 

75  ^f 

9 

.84200 

27 

0.15  800 

9.91  442 

9 

12 

1 

49 
50 

51 

9 

75660 

18 
18 

9 

84227 

27 
27 

26, 

0.15  HZ 

9  91  433 

9 
8 

9 

II 
10 

9 

.2 

9 

8 

08 

I  6 

9 
9 

75678 
75696 

9 
9 

'84280 

0.15  746 
0.15  720 

9.91425 
9  91  416 

52 

9 

75  7M 

9 

.84307 

27 

9.91  407 

9 

8 

•3 

z  7 

24 

53 

9 

•  75  733 

19 

18 
18 
18 
18 
18 
18 
18 

9 

84334 

27 

0.15666 

9.91  398 

9 

7 

4 

3  6 

3  2 

54 
55 

9 

•75  751 

9 

.84361 

27 
27 

0.15639 

9.91  389 

9 
8 

6 

S 

i 

4  5 

5  4 
63 

4  0 
48 

5  6 
64 

9 

•75769 

9 

84388 

0. 15  612 

9.91  381 

50 

9 

.75787 

9 

84415 

27 

0.15585 

9.91  372 

9 

4 

i 

i^ 

9 

.75805 

9 

.84442 

27 

0.15  558 

9  91  363 

9 

3 

l\ 

9 

.75823 

9 

.84469 

27 

0  15  531 

9  91  354 

9 

2 

9 

7^ 

59 

G0_ 

9 

.75841 

9 

.84496 

27 
27 

0.15504 

9  91  345 

9 
9 

0 

9 

75859 

9 

■84523 

0.15477 

9  91  336 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

Prop.  Pts. 

55° 

1 

LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


61 


85^ 


2 

3 

I 

7 
8 

_9_ 
10 
II 

12 
13 

iil 

;i 

17 

18 

i9. 
20 

21 
22 
23 
24 

25 
26 
27 
28 
_29 
30 
31 
32 
33 

^\ 
36 

37 
38 

_39 
40 
41 
42 
43 
44 

J^ 

47 
48 

49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

ii. 
60 


L.  Sin. 


9  75  859 
9  75877 
9  75895 
9  75913 
9  75931 


9  75  949 
9  75  967 
9 -75  985 
9 .  76  003 
9.76  021 


9.76039 
9.76057 
9.76075 
9.76093 
9.76  III 


9.76  129 
9.76  146 
9.76  164 
9.76  182 
9  76  200 


9  76  218 
9.76236 
9  76253 
9  76271 
9  76  289 


9  76  307 
9  76  324 
9.76342 
9.76360 
9.76378 


9  76395 
9  76413 
9  76431 
9.76448 
9 .  76  466 


9.76484 
9.76501 
9.76519 
9  76537 
9  76554 


9.76572 
9.76590 
9 .  76  607 
9 .  76  625 
9 .  76  642 


9 .  76  660 
9.76677 
9.76695 
9.76  712 
9  76  730 


9.76747 

9  76765 
9  76  782 
9 .  76  800 
9.76  817 


9  76835 
9.76852 
9  76  870 
9.76887 
9  76904 


9 .  76  922 


L.  Cos.      d. 


L.  Tang.  c.  d. 


9-84  5£3 
9.84550 
9.84576 
9,84  603 
9.84630 


9.84657 
9  84  684 
9.84  711 
9  84738 
9  84  764 


9.84791 
9.84818 
o .  84  845 
9.84872 
9.84899 


9.84925 
9.84952 
9.84979 
9 .  85  006 
9  85033 


9  85059 
9.85086 

9  85  113 
9.85  140 
9.85  166 


9  85  193 
9  85  220 
9.85247 
9  85273 
9  85  300 


9  85327 
9  85354 
9  85  380 
9  85407 
9  85434 


9 .  85  460 
9.85487 
9  85514 
9  85540 
9  85567 


9  85  594 
9.85  620 
9  85647 
9.85674 
9.85  700 


9,85  727 

9  85  754 
9.85  780 
9  85  807 
9  85834 


9.85  860 
9.85887 
9  85  913 
9  85  940 
9  85  967 


9  85  993 
9  86  020 
9  86  046 
9  86  073 
9  86  100 


9.86  126 


L.  Cotg.  c.  d 


27 
26 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
26 

27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
27 
26 

27 
27 
26 
27 
26 
27 
27 
26 

27 
26 
27 
27 
36 


L.  Cotg. 


0.15477 
0.15450 
C.15  424 
c.  15  397 
0.15370 


15343 
15  316 
15289 
15  262 
15236 


0.15  209 
0.15  182 

o  15  155 
0.15  128 
o.  15  lOI 


o.  15  075 
0.15  048 
o.  15  021 

0.14994 

o.  14  967 


0.14  941 
0.14  914 
0.14887 

o .  14  860 

0.14834 


o.  14  807 
0.14  780 

0.14753 

0.14  727 
o.  14  700 


0.14673 

0.14  646 
o.  14  620 

o  14593 
o.  14  566 

o. 
o. 
o 
o 
o. 


14540 
14  513 

14  486 
14  460 

14433 


o.  14  406 
0.14  380 
o  14353 

0.14326 
0.14300 


L.  Cos. 


0.14273 

o.  14  246 
o.  14  220 

0.14  193 

o.  14  166 


0.14  140 
0.14  113 

o.  14  087 
o.  14  060 

0.14033 


o.  14  007 
o.  13  980 

0.13954 
0.13927 

0.13  900 


[3874 


L.  Tang. 

54° 


91 336 
91 328 

91 319 
91 310 
91 301 


91 292 
91 283 
91 274 
91 266 
91 257 
91 248 
91 239 
91 230 
91 221 
91 212 


91 203 
91 194 
91 185 
91 176 
91 167 


91 158 
91 149 
91 141 
91 132 
91 123 


9  91 114 
9  91 105 
9  91 096 
9.91 087 
9.91 078 


d. 


91  069 
91  060 
91  051 
91  042 
91033 


91  023 

91  014 
91  005 
90  996 
90987 


90978 
90  969 
90  960 

90951 
90942 


90933 
90924 

90915 
90  906 
90  896 


90887 
90878 
90  869 
90  860 
90851 


90  842 
90  832 
90  823 
90  814 
90805 


9.90796 


L.  Sin. 


60 

59 
58 

55 
54 
53 
52 
51 
50 
49 
48 
47 
_46_ 

45 
44 
43 
42 

41 
40 

39 
38 

36 

35 
34 
33 
32 
31 
30 

29 
28 

27 
26 


25 

24 

23 
22 
21 

20" 

19 
18 

17 
16 

15 
14 
13 
12 
II 

To 

9 

8 

7 
6 


Prop.  Pts. 


37 

I 

2.7 

2 

5  4 

3 

8. 1 

4 

10.8 

.  5 

13  5 

.6 

16.2 

•7 

18.9 

.8 

21.6 

9 

243 

26 

2.6 

5-2 

7.8 

10.4 

13.0 

15  6 
18  2 
20,8 
23  4 


18 

1.8 

36 

5  4 

7.2 

9.0 

10  8 

12  6 

14.4 

16.2 


17 
17 
3  4 

\\ 

85 
10.2 
11.9 
136 
15-3 


10 

i.o 
20 

30 
40 

60 
7.0 
8.0 
90 


9 

.1 

0.9 

.2 

1.8 

•3 

27 

•4 

36 

i 

4  5 

5  4 

I 

63 

7  2 

9 

8.1 

8 

08 
1.6 

24 
3  2 

7  2 


Prop.  Pts. 


62 


TABLE  II 


36^ 


9_ 
10 


;i 

i8 

i2_ 
20 

21 
22 

23 
24 


26 
27 
28 

29 

80 

31 

32 
33 
34 

II 
II 

40 

41 
42 

43 
44 

46 

47 
48 

49 
60 

51 

52 
53 
ii 
55 
50 

II 

GO 


L.  Sill. 


76922 
76939 
76957 
76974 
76991 


77009 
77026 

77043 
77061 
77078 


77095 
77  112 
77130 

77147 
77164 


77  181 
77199 
77216 

77233 
77250 


77268 
772S5 
77302 
77319 
77336 


77  353 

77370 
77  3^7 
77405 
77422 


77  439 

77456 
77  473 

77490 
77507 


77  5-4 
77541 
77558 
77  575 
77592 
77G09 
77626 

77643 
77  CGo 

77677 


77694 
77  711 
77728 

77  744 
77761 


77  77^ 
77  795 
77812 
77829 
77846 


77862 
77879 
77896 

77913 
77930 


77946 


I  L.  Cos.  I  <1 


17 

18 

J7 
J7 
18 

»7 
»7 
18 
17 
17 

17 

18 
17 
17 
17 
x8 
17 
17 
17 
18 

17 
17 
17 
17 
17 
17 
17 
18 
17 
17 
17 
17 
17 
17 
«7 
17 
17 
17 
17 
17 
17 
17 
17 
17 


16 


17 


Tangr. 


c.d. 


86 


,86 


126 

o. '53 
,86  179 
86206 
86232 


,86259 
,86285 
86312 
86338 
.86365 


.86392 
1. 86418 
1.86445 
1.86  471 
1.86498 
1.86524 
'•^5551 
1.86577 
1.86603 
86  630 


9.86656 
9.86683 
9.86  709 
9.86736 
9.86  762 


9.86  789 
9.86815 
9.86842 
9.86868 
9.86894 


9.86921 
9.86947 
9.86974 
9.87000 
9.87027 


9-87053 
9.87079 
9.87  106 
9.87  132 
9-87  158 


9.87  185 
9.87  211 
9.87238 
9.87264 
9.87290 


9.87317 

9-87343 
9.87369 
9.87396 
9.87422 


9.87448 

9-87475 
9.87501 
9.87527 
9-87554 
9.87580 
9.87606 

9-87633 
9.87659 
9.87685 


9.8771 


L.  Cotg, 


27 
26 
27 
26 
27 
26 
27 
26 

»7 
27 

26 
27 
26 
27 
26 

27 
26 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
27 
26 
27 
26 

26 
27 
26 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 

27 

26 
26 
27 
26 
26 
27 
26 


26 


L«  Cotg. 


0.13874 
0.13847 
0.13  821 
o.  13  794 
0.13768 


o. 13  741 
0.13  715 
0.13688 
0.13  662 
0-13635 


0.13  608 
0.13582 

0.13555 
0.13529 
0.13502 


0.13476 
0.13449 
0.13423 

0.13397 
0.13370 


13344 
^3317 
13  291 
13264 
13  238 


0.13211 
0.13  185 
0.13  158 
0.13  132 
0.13  106 


0.13  079 
0.13053 
0.13026 
0.13000 
0.12973 


0.12  947 
0.12  921 
0.12  894 
0.12  868 
0.12  842 


0.12  815 
0.12  789 
0.12  762 
0.12  736 
0.12  710 


c.d. 


0.12683 
0.12  657 
0.12  631 
0.12  604 
0.12578 


0.12  552 
0.12  525 
0.12  499 
0.12473 
0.12  446 


0.12  420 
0.12394 
0.12367 
0.12  341 
0.12  315 


[2  289 


L.  Tang. 

53° 


L,  Cos. 


90657 

90648 

90639 

^  ,90  630 

9.90  620 


90  796 
90787 
90777 
90  768 

90759 
90750 
90741 

90731 
90  722 
90713 


90704 
90  694 
90685 
90676 
90667 


90  611 
90  602 
90592 
90583 
90574 


90565 
90555 
90546 
90537 
90527 


90518 
90509 
90499 
90490 
90480 


90471 
90462 
90452 

90443 
90  434 


90424 

90415 

90405 

90396 
90386 


90377 
90368 

90358 
90349 
90339 


90330 
90320 
90  311 
90301 
90292 


90  282 

90273 
90  263 

90254 
90  244 


9  90  235 


L.  Sin. 


GO 

It 
I 

55 
54 
53 
52 

JL 
60 

49 
48 

46 

45 
44 
43 
42 
41 
40 

39 
38 

36 

35 
34 
33 
32 
31 


Prop.  Pis. 


27 

36 

I 

2.7 

26 

•2 

-3 

il 

11 

•4 

10.8 

10.4 

.  1; 

13-5 
16.2 

13.0 

.6 

IS. 6 

•  7 

18. q 

.8 

21.6 

20.8 

.9 

243 

23-4 

18 
1.8 

36 

5  4 
7.2 

9° 
10.8 

12.6 

14.4 
16.2 


17 
17 

3-4 

u 

8.5 

10.2 
II. 9 
136 
15-3 


x6 

1.6 

3  2 
4.8 

6.4 

80 

96 

112 

12  8 

14.4 


10 

.1 

I.O 

.2 

2.0 

•3 
•4 

i 

30 
4.0 

i 

7.0 
8.0 

-9 

9.0 

9 

09 
I 

2 

3 
4 

I 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


63 


37°                                         1 

0 

L.  Sin. 

d. 

L.  Tang. 

c.d.iL.  Cotg.  1 

L.  Cos. 

d. 

60 

Froi».  V\s, 

9  77946 

9  87  711 

0.  12  289 

9  90  235 

I 

9  77963 

9  87  738 

26 

0.12  262 

9 

90225 

S9 

2 

9  77980 

9  87  764 

26 

0.12  236 

9 

90  216 

S8 

^ 

9  77  997 

9  87790 

0.12  210 

9 

90  206 

S7 

27 

,J_ 

9  78013 

'7 

9 

87817 

26 

0.  12  183 

9 

90197 

10 

56 

ss 

.2 

•3 
.4 

9  78030 

9 

87  843 

0.12  157 

9 

90  187 

6 

9  78047 

i5 

9 

87869 

36 

0    12  131 

9 

90178 

S4 

10  8 

7 

9  78  063 

9 

87895 

0    12  105 

9 

90  168 

S3 

,5 

'I  5 
16  2 

8 

9  78080 

9 

87922 

r>(\ 

0.12078 

9 

90  159 

9 

S2 

,6 

9 

9  78097 

[6 

9 

87948 

26 
26 

0.12  052 

9 

90  149 

10 

51 
50 

:l 

18.9 
21.6 

9  78  113 

9 

87974 

0.  12  026 

9 

90139 

1 1 

9 

78  130 

9 

88000 

r%m 

0.12000 

9 

90130 

49 

•9 

24.3 

12 

9 

78  147 

[6 

9 

88027 

on  973 

9 

90  120 

48 

1 

n 

9 

78163 

9 

S°53 

o.ii  947 

9 

90  III 

47 

1 

•4 

9 

78  180 

7 

9 

88079 

26 
26 

o.ii  921 

9 

90  lOI 

10 

46 
4S 

.1 

36 
26 

IS 

9 

78  197 

9 

11  '°5 

O.II  895 

9 

90091 

i6 

9 

78213 

9 

88  131 

O.II  869 

9 

90  082 

9 

44 

.2 

^l 

'7 

9 

78230 

6 

9 

88158 

27 

26 
26 
26 
26 

on  842 

9 

90072 

43 

•3 

78 

i8 

9  78  246 

9 

Z?>  184 

O.II  816 

9 

90063 

9 

42 

•4 

10.4 

19 
20 

9  78  263      , 

7 
6 

9 

88210 

on  790 

9 

90053 

10 

41 
40 

i 

15  6 
18  2 
20  8 
23  4 

9  78280 

9 

88236 

on  764 

9 

90  043 

21 

9 .  78  296 

9 

88262 

on  738 

9 

90034 

9 

3Q 

■  7 
.8 

22 

9 

78313 

9 

88289 

26 
26 
26 
26 

0. II  711 

9 

90024 

38 

2^ 

9 

78329 

9 

88315 

on  685 

9 

90014 

37 

•y 

24 
2S 

9 

78346 

16 

9 

88341 

0  II  659 

9 

90003 

10 

36 

3S 

1 

9 

78362 

9 

88367 

on  633 

9 

8999s 

17 

1-7 

26 

9 

78379 

16 

9 

88393 

on  607 

9 

8998s 

34 

. 

27 

9 

78395 

9 

88420 

27 

26 

O.II  580 

9 

89976 

9 

33 

28 

9 

78412 

^1 
[6 

'7 
[6 

9 

88446 

on  554 

9 

89966 

32 

34 

29 

BO 

9 

78428 

9 

88472 

36 

26 
26 

0.  II  528 

9 

89956 

9 

31 
80 

•3 

•4 

10.2 

9 

78445 

9 

88498 

on  502 

9 

89947 

^i 

9 

78461 

9 

SS524 

oil  476 

9 

89937 

29 

^2 

9 

78478 

9 

88  550 

oil  450 

9 

89927 

28 

i 

II. 9 
136 
IS.:? 

1^ 

9 

78494 

,£. 

9 

88577 

27 
26 
26 

0,11423 

9 

89918 

9 

27 

34 

9 

78510 

17 

t6 

9 

88603 

0  II  397 

9 

89908 

10 

26 

2S 

.q 

9 

78527 

9 

88629 

on  371 

9 

89898 

1 

^6 

9 

78543 

9 

88  6^5 
88681 

26 
26 

O.II  345 

9 

89888 

24 

1 

V 

9 

78560 

[6 
[6 
'7 
[6 

9 

0  II  319 

9 

89879 

•9 

23 

16 

^« 

9 

78576 

9 

88707 

0  II  293 

9 

89869 

22 

^ 

I  6 

39 
40 

9 

78592 

9 

88733 

26 

0  II  267 

9 

89859 

10 

21 

20 

.2 

.3 

U 

9 

78609 

9 

88759 
88786 

0  II  241 

9 

89849 

41 

9 

78625 

9 

27 

0.  II  214 

9 

89840 

9 

19 

.4 

6.4 

42 

9 

78642 

16 
t6 
t7 
[6 
t6 
[6 

9 

88812 

on  188 

9 

89830 

18 

.1; 

8.0 

43 

9 

78658 

9 

88838 

0  II  162 

9 

89820 

17 

.6 

9.6 

44 
4S* 

9 

78674 

9 

88864 

26 
26 

0  II  136 

9 

89810 

9 

16 
IS 

:l 

n.2 

12.8 

9 

78691 

9 

88890 

0.  II  no 

9 

89801 

46 

9 

78707 

9 

88916 

on  084 

9 

89791 

14 

.9 

14.4 

47 

9 

78723 

9 

88942 

26 

O.II  058 

9 

89781 

13 

1 

48 

9 

78739 

9 

88968 

O.II  032 

9 

89771 

12 

1 

49 
50 

_9 
9 

78756 
78772 

16 
[6 

9 

88994 

26 

O.II  006 

9 

89761 

9 

II 
10 

.1 

zo 

I.O 

9 
0.9 

9 

89  020 

0 .  10  980 

9 

89752 

SI 

9 

78788 

9 

89046 

26 

0. 10  954 

9 

89  742 

9 

.2 

2.0 

1.8 

S2 

9 

78805 

I/' 
[6 
16 
6 
6 

9 

89073 

27 

0. 10  927 

9 

89732 

8 

3 

30 

2.7 

S3 

9 

78821 

9 

89099 

0  10  901 

9 

89722 

7 

4 

4.0 

36 

54. 

•>s 

9 
9 

78837 

9 

89125 

26 

26 

0.10875 

9 

89712 

10 

6 
5 

i 

5  0 

6  0 

4.5 

V. 

78  85.3 

9 

89  151 

0.10  849 

9 

89  702 

S6 

9  78  869 

9 

89  1 77 

26 

0.10823 

9 

89693 

9 

4- 

i 

8^o 

S7 

9  78886 

16 
6 
[6 

9 

89  203 

26 

o.io  797 

9 

89  683 

3 

S« 

9.78902 

9 

89  229 

0. 10  771 

9 

89673 

2 

■9 

9.0 

59 
60 

9  78918 

9 

89255 

26 

0.10  745 

9 

89663 

10 

I 
0 

9  78934 

9 

89281 

0. 10  719 

9 

89653 

L.  Cos.      4 

i. 

L.  Cotg. 

c."(i. 

L.  TaiifiT. 

L.  Sin. 

d. 

/ 

Prop.  Fts. 

52^                                           1 

64 


TABLE  II 


38' 


0 

I 

2 

3 
_4 

I 

7 
8 

_9_ 
10 
II 

12 

13 

;i 

17 
i8 

i9_ 
20 

21 
22 
23 

24 

26 

27 
28 
29 


31 
32 

33 

34 

36 
37 
38 
39 
40 

41 
42 

43 
44 

45 
46 

47 
48 

49. 
50 

51 

52 
53 

il 

^^ 

GO 


L.  Sin, 


78934 

78  QW 
78967 
78983 
78999 


79015 
79031 
79047 
79063 
79079 


79095 
79  III 

79  128 

79  144 
79  160 


79176 

79  192 
79  208 
79224 
79240 


79256 
79272 
79288 
79304 
79319 


79  335 
79351 
79367 
79383 
79  399 


79415 
79431 
79  447 
79463 
79478 


79  494 
79510 
79526 
79542 
79558 


79  573 
79589 
79605 
79  621 
79636 


79652 
79668 
79684 
79699 
79715 


79731 
79  746 
79  762 
79778 
79  793 


79809 
79825 
79840 
79856 
79872 


79887 


L.  Cos. 


L.  Taiig. 


c.  d. 


89281 
89307 
89333 
89359 
89385 


89  411 

89437 
89463 
89489 
89515 


89541 
89567 
89593 
89  619 

89645 


89671 
89697 
89723 
89749 
89775 


89801 
89827 
89853 
89879 
89905 


89931 
89957 
89983 
90  009 
90035 


90061 
90  086 
90  112 
90  138 
90  164 


90  190 
90  216 
90  242 
90  268 
90294 


90  320 
90346 
90371 
90397 
90423 


90449 
90475 
90  501 

90527 
90553 


9.90578 

9 .  90  604 

90630 


9.90 
9  90 


656 
682 


90  708 
90734 
90759 
90785 
90  811 


9.90837 


36 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
25 
26 
26 


L.  Cotg.  Ic.  (1 


L.  Cotg. 


o.io  719 
o.ic  693 
o  ic  667 
o  10  64£ 
o.  10  615 


O.IO  589 
o.  10  563 

0.10537 

O.IO  511 
o.  10  485 


0.10459 

0.10433 

o.  10  407 
O.IO  381 

0.10355 


0.10329 

o .  10  303 
O.IO  277 
O.IO  251 
o.  10  225 


o.  10  199 
O.IO  173 
O.IO  147 

O.  10  121 

0.10095 


O.IO  069 
o.  10  043 
O.IOOI7 
0.09  991 
o  •  09  965 


0.09939 

0.09  914 

0.09888 

o .  09  862 
0.09  836 


0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 


o .  09  680 
0.09  654 
o .  09  629 
o .  09  603 

0.09577 


0.09551 

0.09  525 

0.09499 
0.09473 
0.09447 


o .  09  422 
0.09  396 
0.09  370 

0.09344 

0.09  318 


0.09  292 
0.09  266 
o .  09  241 
o  09  2 1 5 
o  09  189 
0.09  163 


L.  Tang. 

5r 


L.  Cos. 


89653 
89643 
89633 
89  624 
89  614 


89  604 
89594 
89584 
89574 
89564 


89554 
89544 
89534 
89524 
89514 


89504 
89495 
89485 

89475 
89465 


89455 
89445 
89435 
89425 

89415 


89405 
89395 
89385 
89375 
89364 


89354 
89344 
89334 
89324 

89314 


89304 
89294 
89284 

89274 
89  264 


89254 
89244 
89233 
89  223 
89213 


89203 

89193 

8918:5 

89173 

89  Ib2 


89152 
89  142 

89  132 
89  122 
89  112 


89  lOI 

89  091 

89081 
89071 
89  060 


9 .  89  050 


L.  Sin. 


60 

59 
58 
57 

55 
54 
53 
52 
51 
50 
49 
48 
47 
_46_ 

45 
44 
43 
42 
41 
40 

39 
38 

36 

35 
34 
33 
32 

Jl 
30 

29 
28 

27 
26 


Prop.  Vis. 


26 

I 

26 

2 

S-2 

3 

7.8 

4 

10.4 

5 

13   0 

6 

15.6 

7 

18   2 

8 

20.8 

9 

23  4 

17 

2 

1  ■ 

3 

3 

4 

6 

8. 

.6 

10. 

.7 

II 

.8 

13 

9 

15 

25 

25 

50 

7  5 
10  o 

12  5 
15  o 

17  5 
20.0 
22.5 


16 

I 

1.6 

2 

32 

3 

4.8 

4 

S4 

.1^ 

8  ol 

.6 

96 

•  7 

II. 2 

.8 

12.8 

9 

14-4 

XI 

I 

I. 

2 

2 

3 

3 

•4 

4 

I 

7 

I 

7 

.8 
•9 

8 
9 

zo 

9 

10 

0 

20 

I 

30 

2 

4.0 

3 

SO 

4 

6.0 

S 

7.0 

6 

8.0 

I 

90 

15 

30 

6.0 

7  5 
9.0 

10. 5 
12.0 

^3  5 


Prop.  Pis. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


65 


39°                                         1 

t 

0 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

60 

Prop. 

Pts. 

9  79887 

16 

9.90837 

26 

0.09  163 

9  89  050 

■       1 

I 

9 

79903 

TC 

9 

90«b3 

26 

0.09  137 

9 

89  040 

S9 

1 

2 

3 
_4_ 

9 
9 
9 

79918 
79  934 
79950 

16 
16 
15 
16 

9 
9 
9 

90889 
90914 
90940 

2S 
26 
26 
26 

0.09  III 
0,09086 
0 .  09  060 

9 
9 
9 

89  030 
89  020 
89009 

10 
II 
10 

58 

.1 

.2 
3 
4 

26 
2.6 

li 

10.4 

ill 

208 

9 

79965 

9 

90966 

0.09034 

9 

88999 

6 

9 

79981 

15 
16 

9 

90992 

oA 

0.09008 

9 

88989 

54 

7 

9 

79996 

9 

91  018 

0.08982 

9 

88978 

S^ 

8 

9 

80012 

15 
16 

16 

9 

91043 

26 
26 

0.08957 

9 

88968 

52 

9 
10 

9 

80027 

9 

91  069 

0.08  931 

9 

88958 

10 

51 
50 

I 

9 

80043 

9 

91095 

0.08905 

9 

88948 

II 

9 

80058 

9 

91  121 

26 

0.08879 

9 

88937 

49 

9 

23  4 

12 

9 

80074 

9 

91  147 

0.08853 

9 

88927 

48 

n 

9 

80089 

16 

9 

91  172 

25 

26 
26 
26 
26 

0.08828 

9 

88917 

47 

1 

14 
15 

9 

80  105 

IS 
16 

9 

91  198 

0.08802 

9 

88906 

10 

46 
45 

25 

25 

9 

80  120 

9 

91  224 

0.08  776 

9 

88896 

16 

9 

80  136 

9 

91  250 

0.08  750 

9 

88  886 

44 

.2 

50 

17 

9 

80  151 

9 

91  276 

0.08  724 

9 

88875 

43 

•3 

7-5 

18 

9 

80166 

16 

9 

91  301 

25 

26 
26 
26 

0.08  699 

9 

88865 

42 

•4 

10.0 

19 
20 

9 

80  182 

15 

16 

9 

91  327 

0.08  673 

9 

88855 

II 

41 
40 

i 

12.5 
15  0 

9 

80197 

9 

91353 

0.08  647 

9 

88844 

21 

9 

80213 

9 

91  379 

0.08  621 

9 

88834 

39 

i 

17  5 

22 

9 

80228 

16 

9 

91404 

25 

26 
26 
26 

0.08  596 

9 

88824 

10 

38 

20.0 

2S 

9 

80244 

9 

91430 

0.08  570 

9 

88813 

37 

•9 

22.5 

24 
2"^ 

^9 
"9 

80  259 

15 

16 

9 

91456 

0.08  544 

9 

88803 

10 

36 

35 

1 

80274 

9 

91482 

0.08518 

9 

88793 

16 

26 

9 

80290 

9 

91  507 

26 
26 
26 
25 
26 
26 
26 

0.08493 

9 

88782 

34 

1.6 

27 

9 

80305 

9 

91  533 

0.08467 

9 

88772 

33 

28 

9 

80320 

16 
15 

9 

91  559 

0.08  441 

9 

88761 

32 

8  0 

29 

30 

9 

8033^ 

9 

91585 

0.08  415 

9 

88751 

10 
10 

31 
80 

■3 
•4 

9 

80351 

9 

91  610 

0.08  390 

9 

88741 

31 

9 

80366 

16 

9 

91  636 

0.08  364 

9 

88730 

11 

9.6 
112 

32 

9 

80382 

9 

91  662 

0.08338 

9 

88  720 

10 

28 

i 

33 

9 

80397 

9 

91688 

0  08  312 

9 

88  709 

II 

27 

12  8 

34 

3S 

9 

80412 

16 

9 

91  713 

25 

26 
26 
26 

0.08287 

9 

88699 

10 
II 

26 
25 

.g 

9    .  . 

lA   A 

9 

80428 

9 

91  739 

0.08  261 

9 

88  688 

36 

9 

80443 

9 

91  765 

0.08  235 

9 

88678 

10 

24 

1 

37 

9 

80458 

9 

91  791 

0  08  209 

9 

88  668 

10 

23 

X5 

IS 

30 

38 

9 

80473 

IS 

16 
15 

9 

91  816 

25 

0.08  184 

9 

88  657 

II 

22 

39 
40 

9 

80489 

9 

91842 

26 

0.08  158 

9 

88  647 

10 
11 

21 
20 

.2 

3 
.4 

9 

80504 

9 

91  868 

0  08  132 

9 

88636 

41 

9 

80519 

15 

9 

91  893 

25 
26 
26 
26 

2S 
26 

0.08  107 

9 

88626 

10 

IQ 

42 

9 

80534 

15 

9 

91  919 

0.08081 

9 

88615 

II 

18 

7  5 

43 

9 

80550 

9 

91  945 

0.08  055 

9 

88605 

17 

5 

90 

44 
4S 

9 

80565 

IS 

9 

91  971 

0 .  08  029 

9 

88594 

10 

16 
15 

i 

10.5 
12  0 

9 

80580 

9 

91  996 

0 .  08  004 

9 

88584 

46 

9 

80595 

*s 

9 

92  022 

0.07978 

9 

88  573 

II 

14 

9 

13  5 

47 

9 

80610 

IS 

9 

92  048 

0.07952 

9 

88  S63 

10 

n 

1 

48 

9 

80625 

IS 

16 

^5 

9 

92073 

25 
26 
26 

0.07927 

9 

88  552 

II 

12 

1 

49 
60 

9 

80641 

9 

92099 

0.07  901 

9 

88542 

II 

II 
10 

.1 

II 
I .  I 

10 
1.0 

9 

80656 

9 

92  125 

0.07875 

9 

88531 

SI 

9 

80671 

^s 

9 

92  150 

2S 

0.07850 

9 

88521 

10 

p 

.2 

2.2 

2.0 

52 

9 

80686 

IS 

9 

92176 

26 

0.07  824 

9 

88510 

II 

8 

3 

3  3 

30 

S3 

9 

80  701 

9 

92  202 

0.07  798 

9 

88499 

I' 

7 

4 

4  4 

4.0 

54 

SS 

9 

80  716 

15 
15 

9 

92227 

2S 
26 

0.07  773 

9 

88489 

10 
II 

6 

S 

I 

U 

50 
6.C 

9 

80731 

9 

92253 

0.07  747 

9 

88478 

S6 

9  80  746 

15 

16 

9 

92279 

0.07  721 

9 

88468 

10 

4 

:i 

ii 

7.0 
8.C 

S7 

9  80  762 

9 

92304 

2S 

0.07  696 

Q 

88457 

IX 

3 

S8 

9  80  777 

15 

9 

92330 

0.07  670 

9 

88447 

10 

2 

•9 

9  9 

9.0 

59 

9  80792 

15 
15 

9 
9 

92356 
92381 

2S 

0.07  644 

9  88  436 

II 
II 

I 
0 

9  80  807 

0.07  619 

9.88425 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

f 

Prop.  Pis. 

50^                                         1 

66 


TABLE  II 


40°                                         1 

t 

L.  Sin. 

d. 

L.  Tang. 

c.  d. 

L.  Cotg. 

L.  Cos. 

d. 

w 

Prop.  Pts. 

0 

9.80807 

9.92381 

0.07  619 

9.88425 

I 

9 

80  822 

9 

92407 

0.07593 

9 

88415 

S9 

2 

9 

80837 

9 

92433 

0.07567 

9 

88404 

S8 

26 

3 

9 

80852 

9 

92458 

25 

0.07  542 

9 

88  394 

S7 

T 

2  6 

4 

9 

80867 

9 

92484 

26 

0.07  516 

9 

88383 

II 

56 

55 

2 

3 

'7s 

9 

80882 

9 

92510 

0.07490 

9 

88  372 

6 

9 

80  897 

9 

92535 

25 

0.07465 

9 

88362 

54 

4 

10  4 

7 

9 

80  912 

9 

92561 

26 

0.07439 

9- 

88351 

53 

13  0 

8 

9 

80927 

9 

92587 

0.07413 

9. 

88340 

52 

(5 

15  6 

9 
10 

9 

80942 

9 

92  612 

25 
26 

0.07388 

9 

88330 

II 

51 
50 

7 
8 

182 
20  8 

9 

80957 

9 

92  638 

0.07362 

9 

88319 

II 

9 

80972 

9 

92  663 

25 
26 
26 

0.07337 

9 

88308 

49 

9 

23  4 

12 

9 

80987 

9 

92689 

0.07  311 

9 

88298 

48 

1 

n 

9 

81  002 

9 

92  715 

0.07  285 

9- 

88287 

47 

1 

14 

9  81  017 

9 

92  740 

26 
26 

0.07  260 

9 

88276 

10 

46 
45 

25 

25 

9  81  032 

9 

92766 

0.07234 

9 

88266 

1 

i6 

9 

81  047 

9 

92  792 

0.07  208 

9 

88255 

44 

2 

50 

17 

9 

81  061 

9 

92817 

25 
26 

0.07 183 

9. 

88244 

43 

3 

7  5 

i8 

9 

81  076 

9 

92843 

0.07 157 

9 

88234 

42 

4 

10  0 

19 
20" 

9 

81  091 

9 

92868 

25 
26 

26 

0.07132 

9 

88223 

II 

41 
40 

I 

12.5 
15  0 

17  5 
20  0 

9 

81  106 

9 

92894 

0.07  106 

9 

88212 

21 

9 

81  121 

9 

92  920 

0.07  080 

9 

88201 

39 

22 

9 

81  136 

9 

92945 

25 
26 

0.07055 

9 

88  191 

° 

38 

n 

22.5 

2^ 

9 

81  151 

9 

92971 

0.07029 

9 

88180 

37 

24 
2S 

9 

81  166 

9 

92996 

25 

26 

26 

0.07004 

9 

88  169 

II 

36 
35 

1 

9 

81  180 

9 

93022 

0.06978 

9 

88158 

IS 

26 

9 

81  195 

9 

93048 

0.06952 

9 

88  148 

34 

^ 

15 

3  0 

4  5 

6  0 

7  5 

27 

9 

81  210 

9 

93073 

25 

26 

0.06927 

9 

88137 

33 

2 

28 

9 

81  225 

9 

93099 

0  06  901 

9 

88  126 

32 

•3 
•4 

•  S 

29 

30 

_9 
9 

81  240 
81  254 

9 

93  124 

25 
26 

0  06876 

9 

88  115 

10 

31 
30 

9 

93  150 

0.06  850 

9 

%%  105 

SI 

9 

81  269 

9 

93  175 

25 

0.06825 

9 

88094 

29. 

.6 

90 

32 

9 

81  284 

9 

93201 

26 

0.06  799 

9 

88083 

28 

.7 

10  5 

33 

9 

81  299 

9 

93227 

0.06  773 
0  06  748 

9 

88072 

27 

.8 

12  0 

34 
3S 

9 

81  314 

9 

93252 

25 

26 

9 

88061 

10 

26 
25 

9 

13  5 

9 

81  328 

9 

93278 

0.06  722 

9 

88051 

1 

3^ 

9 

81343 

9 

93303 

25 

0  06  697 

9 

88040 

24 

1 

37 

9 

81358 

9 

93329 

0  06  671 

9 

88029 

23 

14 

3« 

9 

81372 

9 

93  354 

25 

0  06  646 

9 

88018 

22 

I 

14 
2  8 

4  2 

39 
40 

9 

,81  387 

9 

93380 

26 

0  06  620 

9 

88007 

11 

21 

20 

.2 

.3 

9  81  402 

9 

93406 

0.06594 

9 

87996 

A\ 

9,81  417 

9 

93431 

25 

0.06  569 

9 

87985 

19 

•  4 

56 

42 

9  81  431 

9 

93  457 

26 

0 .  06  543 

9 

87975 

18 

.5 

7  0 

43 

9  81  446 

9 

93482 

25 

0  06  518 

9 

87964 

17 

.6 

84 

44 
4S 

9  81  461 

9 

93508 

26 

25 

0  06  492 

9 

87953 

II 

16 

^5 

•7 
.8 

98 
II  2 

9^1  475 

9 

93  533 

0.06467 

9 

87942 

46 

9  81  490 

9 

93  559 

26 

0.06  441 

9 

87931 

14 

9 

12  6 

47 

9  81  505 

9 

93584 

25 

0  06  416 

9  87  920 

13 

1 

48 

9  81  519 

9 

93  610 

0.06390 

9  87  909 

12 

1 

49 
50 

9  «i  534 

9 

93636 

26 

25 

0  06  364 

9  87  898 

II 

II 
10 

I 

XI 

II 

10 

ID 

9  81  549 

9 

93661 

0  06339 

9  87  887 

.S» 

9  8'  563 
9  «•  578 

9 

93687 

26 

0  06313 

9  87877 

9 

.2 

22 

2  0 

S2 

9 

93712 

25 

0.06288 

9 

87866 

8 

3 

3  3 

3  0 

S3 

9  81  592 

9 

93  738 

26 

0.06  262 

9 

87855 

7 

4 

4  4 

4  0 

54 

9  81  607 

9 

93  763 

25 
26 

0.06  237 

9 

87844 

6 

5 

.^5 

IS 

5S 

9  81  622 

9 

93  789 

0.06  211 

9 

^l^ZZ 

5 

6 

6  6 

5^ 

9  81  636 

9 

93814 

25 

0.06  186 

9 

87822 

4 

.8 

1^ 

^0 

^s^ 

9  81  651 

9 

93  840 

26 

0.06  160 

9  87  811 

3 

9  81  665 

9 

93865 

25 

0.06  135 

9.87800 

2 

y 

9  9 

90 

59 
60 

9  81  680 

9 

93891 

26 
25 

0.06  109 

9  87789 

II 

0 

9  81  694 

9 

93916 

0  06  084 

9  87  778 

L.  Cos. 

L.  Cotg. 

c.  d. 

L.  Tang. 

L.  Sin. 

f 

Prop.  Pt8. 

49^                                            1 

LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


67 


41 


9_ 
10 
II 

12 

\i 

i8 
20 

21 

22 
23 

24 

25 
26 
27 
28 

f9 
30 

31 
32 
33 
Jl 

36 
37 
38 
39 
40 
41 
42 
43 
44 

46 

47 
48 

49 

50 

51 

52 
53 
_54 
55 
5^ 

00 


L.  Sin, 


694 
709 
723 
738 

752 


767 
781 
796 
810 
825 


839 
854 
868 
882 
897 


911 
926 
940 

955 
969 


983 
998 
82  012 
82  026 
82  041 


82055 
82  069 
82084 
82098 
82  112 


82  126 
82  141 
82155 
82  169 
82  184 


82  198 
82  212 
82  226 
82  240 
82255 


82  269 
82283 
82  297 
82  311 
82_326_ 

82340 
82354 
82368 
82382 
82396 


82  410 
82  424 
82439 

82453 
82467 


82481 
82495 
82  509 
82523 
82537 


82551 


L.  Cos. 


d. 


Tang. 


93916 
93942 
93  967 

93  993 

94  018 


94044 
94  069 
94095 
94  120 
94  146 


94  171 
94  197 
94  222 
94248 
94273 


94299 
94324 
94350 
94  375 
94401 


94  426 
94452 
94  477 
94503 
94528 


94  554 
94  579 
94  604 
94630 
94655 


94  681 
94  706 
94  732 
94  757 
94783 


94808 
94834 
94859 
94884 
94910 


94  935 
94961 

94  986 

95  012 
95037 


95  062 
95088 
95  113 
95  139 
95  164 


95  190 
95215 
95  240 
95  266 
95291 


95317 
95  342 
95368 
95  393 
95  418 


c.  d. 


95  444 


Cotff. 


c.  d. 


L.  Cotg. 


o .  06  084 
0.06  058 
0.06033 
0.06  007 
0.05  982 


0.05  956 
0.05931 
0.05  905 
0.05  880 
0.05  854 


0.05  829 
0.05  803 
o  05  778 
o  05  752 
0.05  727 


o  05  701 
o  05  676 
0.05  650 
0.05  625 
0.05  599 


0.05  574 
0.05  548 
0.05  523 
0.05  497 
0.05  472 


0.05  446 
0.05  421 
o  05  396 
0.05  370 
005  345 


0.05  319 
0.05  294 
0.05  268 
0.05  243 
6.05  217 


0.05  192 
0.05  166 
0.05  141 
o  05  116 
o .  05  090 


0.05  065 
o  05  039 
o  05  014 
o  04  988 
o  04963 


o  04  938 
o  04  912 
o  04  887 
0.04  861 
o .  04  836 


0.04  5IO 

0.04  785 
o  04  760 
o  04  734 
0.04  709 


o .  04  683 
o .  04  658 
o .  04  632 
0.04  607 
o .  04  582 


0.04  556 


L.  Tang. 

48° 


L.  Cos. 


87778 
87767 
87756 
87745 
87734 


87723 
87712 
87  701 
87  69c 
87679 


87668 

87657 
87646 

87635 
87624 


87613 
87601 
87590 
87579 
87  568 


87557 
87546 
87535 
87524 
87513 


87501 
87490 
87479 
87468 

87457 


87446 
87434 
87423 
87  412 
87  401 


87390 
87378 
87367 
87356 


9  87  345 


9  87  334 
9  87322 

9  873" 
9  87300 
9.87288 


87277 
87266 
87255 
87243 
87232 


87221 

87  209 

87  198 

9  87  187 

9  87  175 


9  87  164 
9  87  153 
9  87  141 
9  87  130 
9  87  119 


9.87  107 


L.  Sin, 


d. 


«0 

59 
58 
57 
56 


55 
54 
53 
52 
_5L 
50 
49 
48 
47 
46 


45 
44 
43 
42 
41 
40 

39 
38 

36 

35 
34 
33 
32 
31 


25 
24 
23 
22 
21 

20 

^9 
18 

17 
16 


15 
14 
13 
12 
II 

9 
8 

7 
6 


Prop.  Pts. 


6 
2 

8 

4 
o 

18.2 
20.8 
23  4 


25 

25 
50 

7  5 
10.0 

12.5 
15  o 

17  5 
20.0 
22.5 


15 

IS 

30 

6.0 

7-5 
9.0 

10.5 
12.0 

13  5 


X4 

14 
8 
2 
6 
o 

4 
8 
2 
6 


13 

.1 

12 

.2 

2.4 

3 

36 

•  4 

48 

6.0 

.6 

7.2 

•  7 

8.4 

.8 

9.6 

•9 

0.8 

Prop.  Pts. 


68 


TABLE  II 


42^ 


2 

3 

A 

5 
6 

7 
8 

9^ 

10 

II 

12 

13 
14 


11 

17 

i8 

i9_ 
20 

21 

22 
23 
24 

26 
27 
28 
29 

30 

31 
32 
33 
34_ 

36 

37 
38 
39 
40 

41 
42 

43 
44 

45 
46 

47 
48 

49 

50 

51 

52 
53 

I 
II 

60 


L.  Sin. 


82551 
82565 
82  579 
82  593 
82  607 


82621 
82  635 
82  649 
82663 
82677 


82  69_i 
82  705 
82  719 
82  733 
82  747 


82  76_i 

82775 
82788 
82802 
82816 


82830 
82844 
82858 
82872 
8288=: 


82899 
82913 
82  927 
82  941 
82955 


82968 
82982 

82  996 

83  010 
83023 


83037 
83051 
83065 
83078 
83092 


83  106 
83  120 
83133 
83  147 
83  161 


83  174 
83  188 
83  202 
83215 
83  229 


83242 
83256 
83  270 
83283 
83297 


83310 
83324 
83338 
83351 
83365 


983378 


Cos. 


d.     L.  Tang. 


95  444 
95  469 
95  495 
95  520 
95  545 


95  571 
95596 
95  622 
95647 
95  672 


95698 
95  723 
95  748 
95  774 
95  799 


95  825 
95850 
95875 
95901 
95  926 


95952 

95  977 

96  002 
96  028 
96053 


96  078 
96  104 
96  129 

96155 
96  180 


96  205 
96231 
96  256 
96281 
96307 


96332 
96357 
96383 
96  408 

96433 


96459 
96  484 
96  510 

96535 
96  560 


96586 
96  611 
96636 
96  662 
96687 


96  712 
96738 
96  763 
96788 
96  814 


96839 
96864 
96  890 

96915 
96  940 


c.d, 


96  966 


d.  I  L.  Cotg. 


25 
26 
25 
25 
26 

25 
26 

25 

25 
26 

25 
25 
26 

25 
26 

25 
25 
26 

25 
26 

25 
25 
26 

25 
25 
26 
25 
26 
25 
25 
26 

25 

25 
26 
25 

25 
26 
25 
25 
26 

25 
26 
25 
25 
26 

25 
25 
26 
25 

25 

26 
25 
25 
26 
25 

25 
26 
25 
25 
26 


c.d. 


L.  Cotg. 

0.04  556 
o  04531 
o .  04  505 
o  04  480 
o  04  455 


o  04  429 
o  04  404 

0,04378 
0.04353 

0.04  328 


o .  04  302 
0.04  277 
o  04  252 
o .  04  226 
0.04  201 


0.04  175 

0.04  150 
0.04  125 
o .  04  099 
o .  04  074 
o .  04  048 
o .  04  023 

0.03  998 

0.03  972 

0.03947 


0.03  922 
0.03  896 

0.03871 

0.03  845 
0.03  820 


0.03  795 
0.03  769 
0.03  744 
0.03  719 
0.03  693 


0.03  668 
0.03643 
o  03617 
0.03  592 
0.03  567 


0.03541 
0.03  516 
o .  03  490 
o .  03  465 

0.03440 


0.03  414 
0.03389 
0.03364 
0.03338 
0.03313 


0.03  288 

0.03  262 

0.03237 

0.03  212 
0.03  186 


0.03   161 

o  03  136 
o  03  no 
o  03  085 
o ,  03  060 


0.03034 


L.  Tang. 

470 


I 

.  Cos. 

d 

• 

60 

9,87 107 

9 

87096 

, 

59 

9 

87085 

58 

9 

87073 

57 

_9_ 
9 

87062 

2 

56 

55 

87  050 

9 

87039 

54 

9 

87028 

5^ 

9 

87016 

52 

9 

87005 

2 

51 
50 

9 

86993 

9 

86982 

^ 

49 

9 

86970 

48 

9 

86959 

47 

9 

86947 

I 

46 
45 

9 

86936 

9 

86924 

44 

9 

86913 

43 

9 

86902 

42 

9 

86890 

I 

41 
40 

9 

86879 

9 

86867 

39 

9 

86  855 

38 

q 

86844 

37 

9 

86832 

^ 

36_ 

35 

9 

86821 

9 

86809 

M 

9 

86798 

^ 

S3 

9 

86786 

^2 

9 

86775 

2 

31 
30 

9 

86763 

9 

86752 

' 

29- 

9 

86  740 

28 

9 

86728 

27 

9 

86717 

2 

26 

25 

9 

86705 

9 

86694 

' 

24 

9 

86682 

23 

9 

86670 

22 

9 

86659 

2 

21 
20 

9 

86647 

9 

86  63s 

19 

9 

86624 

18 

9 

86612 

17 

9 

86600 

I 

16 
15 

9 

86589 

9 

86577 

14 

9 

86  565 

13 

9 

86554 

12 

9 

86542 

2 

II 
10 

9 

86530 

9 

86518 

9 

9 

86507 

8 

9 

86495 

7 

9 

86483 

6 

5 

9 

86472 

9 

86460 

4 

9 

86448 

3 

9 

86436 

2 

9  86  425 

0 

9  86413 

I 

..  Sin. 

i 

. 

/ 

Prop.  Pts, 


36 

2  6 


25 

25 

o 

5 


5 

7 

10.0 

12.5 


14 

14 

2.8 


13 

.1 

1 . 

.2 

2 

•3 

3 

•  4 

5 

.5 

6. 

6 

7 

7 

9 

.8 

10. 

9 

II . 

12 

.1 

I  2 

.2 
3 

•4 

3I 

4.8 
6.0 

9 

II 
I 

2.2 

3  3 

4  4 

II 

7  7 

8  8 

9  9 


Prop.  Pts. 


LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS 


69 


43^ 


L.  Sin. 


7 
8 

9_ 
10 
II 

12 

13 

lA 
15 

i6 

17 
i8 

i9_ 
20 

21 
22 
23 

24 

26 
27 
28 
29 


31 
32 

33 
34 

36 

37 
38 
J9 
40 

41 
42 
43 
_44_ 

46 

47 
48 

49 
50 

51 

52 
53 
id 
55 
56 
57 
58 
i9_ 
60 


983378 
9  83392 
9  83405 
9.83419 

9  83432 


9.83446 
9  83  459 
9  83473 
9  83  486 
9  83  500 


9  83513 
9  83  527 
9  83540 


83554 
83567 


83581 

83594 

83608 

83621 

83,634^ 

83648 

83661 

83674 
83688 
83701 


83715 
83728 

83741 
83755 
83768 


83781 

83795 
83808 
83821 
83834 


83848 
83861 

83874 
83887 
83901 


83914 
83927 

83940 
83954 
83.967 
83 
83993 
84  006 
84  020 
84033 


9 .  84  046 
9  84  059 
9  84  072 
9  84  085 
9  84  098 


84  112 
84  125 
84138 
84  151 
84  164 


9  84177 


L.  Cos. 


L.  Tang. 


9  96  966 

9  96  991 
9  97  016 
9  97042 
9.97067 


97092 
97  118 

97  143 
97168 

97  193 


97219 
97244 
97269 

97295 
97320 


97  345 
97371 
97396 
97421 
97  447 


97472 
97  497 
97523 
97548 
97  573 


97598 
97624 
97649 
97674 
97  700 


97725 
97750 
97776 
97  801 
97  826 


97851 
97877 
97902 
97927 
97  953 


97978 
98  003 
98  029 
98054 
98079 


98  104 
98  130 
98  155 
98  180 
98  206 


9  98  231 
9  98  256 
9  98  281 
9  98307 
9  98  332 


9  98357 
998383 
9 .  98  408 

9  98433 
9.98458 


9.98484 


L.  Cotg.  c.  (I 


c.  d. 


L.  Cotg. 


0.03034 
o  03  009 
o .  02  984 
0.02  958 

0.02933 


0.02  908 
o  02  882 
0.02  857 
0.02  832 
o  02  807 


02  781 
02  756 
02  731 
02  705 
02  680 
02  655 
02  629 
02  604 
0.02  579 
0.02  553 


0.02  528 
0.02  503 
0.02  477 

o .  02  452 
0.02  427 


o .  02  402 
0.02  376 
o  02  351 
o .  02  326 
o  02  300 


o  02  275 
o .  02  250 
0.02  224 
o  02  199 
0.02  174 


o  02  149 
o  02  123 
o .  02  098 
o .  02  073 
0.02  047 


o .  02  022 
o.oi  997 
o.oi  971 
o.oi  946 
o.oi  921 


O.OI  896 

O.OI  870 
0.01  845 
O.OI  820 
O.OI  794 


O.OI  769 
O.OI  744 
O.OI  719 
O.OI  693 
O.OI  668 


O.OI  643 
o  01  617 
o  01  592 
O.OI  567 
O.OI  542 


O.OI  516 


L.  Tang. 

46° 


L 

.  Cos. 

d. 

9  86413 

60" 

9 

86401 

59 

9 
9 

86389 
86377 

2 

58 

57 

9 

86366 

2 
2 

56 

55 
S4 

9 
9 

86354 
86342 

9 

86330 

■^^ 

9 

86318 

'^2 

9 

86306 

51 
50 

9 

86295 

9 

86283 

49 

9 

86271 

48 

9 

86259 

47 

9 

86247 

2 

46 

45 

9 

86235 

9 

8b  223 

44 

9 

86  211 

43 

9 

86200 

42 

9 

86188 

2 

41 
40 

9 

86176 

9 

86  164 

i:^ 

39 

9 

86152 

38 

9 

86  140 

37 

9 

86128 

[2 

36 

35 

9 

86  116 

9 

86  104 

34 

9 

86092 

33 

9 

86080 

32 

9 

86068 

[2 

31 
30 

9 

86056 

9 

86044 

29 

9 

86032 

28 

9 

86020 

27 

9 

86008 

12 

26 
25 

9 

85996 

9 

85984 

24 

9 

85972 

23 

9 

85  960 

22 

9 

85948 

C2 

21 
20 

9 

85936 

9 

85924 

19 

9 

85  912 

18 

9 

85900 

17 

9 

85888 

[2 

16 
IS 

9 

85876 

9 

85  864 

14 

9 

85851 

3 

13 

9 

85839 

12 

9 

85827 

2 

II 
10 

9 

85815 

9 

85  803 

9 

9 

85  791 

8 

9 

85  779 

7 

■9 

85  766 

3 

2 

6 

5 

9 

85754 

9 

85  742 

2 

4 

9 

85  730 

3 

9 

85718 

2 

.    2 

9 

85706 

2 

I 
0 

9  85693 

13 

L.  Sin. 

d. 

/ 

Prop.  Pts. 


26 

I 

2. 

2 

5 

3 

7- 

4 

10 

13 

6 

7 

18 

8 

20 

9 

23 

25 

I 

25 

2 

50 

3 

7-5 

4 

10.0 

5 

12  s 

6 

15  0 

7 

175 

8 

20.0 

9 

22.5 

14 

,1 

1-4 

.2 

2.8 

•3 

4.2 

•4 

5-6 

•5 

7.0 

.6 

.8.4 

•7 

98 

.8 

II. 2 

•9 

12.6 

13 

I 

I . 

2 

2 

3 

3 

4 

5 . 

6. 

6 

7- 

7 

9 

8 

10 

9 

II. 

12 

II 

.1 

1.2 

I. 

.2 

2.4 

2. 

3 

36 

3 

•4 

4.8 

4- 

5 

6.0 

5- 

6 

7.2 

6. 

7 

8.4 

7- 

8 

9.6 

8. 

9 

10.8 

9- 

Prop.  Pts. 


70 


TABLE  II 


44°                                        1 

/ 

L.  Sin. 

d. 

L.  Tang. 

C.d. 

L.  Cotg. 

L.  Cos.     d 

I. 

60 

Prop.  Pts. 

0 

9  84  177 

9.98484 

0.01  516 

985693      , 

I 

9  84  190 

13 

9  98509 

25 

0.01  491 

9  85  681 

SP 

2 

9 

84203 
84216 

*3 

9  98534 

25 

0.01  466 

985669 

58 

9fi 

S 

9 

9  98  560 

0.01  440 

9  85  657 

S7 

, 

2  6 

78 

4 

s 

9 

84229 

»3 
*3 

998585 

25 

0.01  415 

9.85645   ; 

3 

56 

55 

2 

3 
.4 

9 

84242 

9  98  610 

001  390 

985632 

6 

9 

84  2.S5 

»3 

9  98635 

0  01  365 

9  85  620 

54 

10    4 

7 

9 

84269 

9  98  661 

0  01339 

9.85608   ' 

5S 

I 

IT,    0 

8 

9 

84  282 

13 

9  98686 

0  01  314 

985596   ; 

52 

15  6 

9 
10 

9 

84295 

13 
13 

9  98  711 

26 

0.01  289 

985583   ; 

J 

2 

51 
50 

7 
8 

18  2 
20  8 

9  84  308 

9  98737 

0  01  263 

985571 

II 

9  84  321 

»3 

9  98  762 

0  01  238 

985559   , 

49 

9 

23  4 

12 

9  84  334 

9  98  787 

0  01  213 
0  01  188 

985547   , 

48 

1 

IS 

9  84  347 

9  98812 

26 
25 

985534  ■ 

47 

1 

14 
IS 

9  84  360 

13 
13 

_9_ 
9 

98838 
9886^ 
98888 

0.01  162 

9.85522    ; 

2 

46 
45 

.1 

25 

2  5 

9  84  373 

0.01  137 

985510 

i6 

9  84385 
9.84398 

9 

0.01  112 

985497 

3 

44 

.2 

5.0 

17 

13 

9 

98913 

25 
26 

0.01  087 

9.85485 

43 

•3 

7  5 

i8 

9  84  411 

13 

9 

98939 

0  01  061 

985473    , 

42 

•4 

10  0 

19 
20 

9  84424 

13 
13 

9 

98964 

25 
25 
26 

0  01  036 

9  85  460 

3 

3 

41 
40 

■  i 

.7 
.8 

•9 

12  5 
15  0 

17  5 
20.0 
22  5 

9  84  437 

9 

98989 

o.oi  on 

985448    , 

21 

9  84  450 

13 

9 

99015 

0.00  985 

985436    , 

39 

22 

9.84463 
9.84476 

9 

99040 

0 .  00  960 

985423    , 

3 

38 

23 

9 

99065 

0.00935 

9-85411 

37 

24 

2S 

9  84  489 

13 

9 

99090 

26 

0.00910 

9  85  399    , 

3 

36 

35 

1 

9  84  502 

9 

99  116 

0.00  884 

9.85386 

14 

26 

984515 

*3 

9 

99  141 

25 

0.00859 

985374    , 

34 

.1 

27 

9 

84528 

>3 

9 

99  166 

25 

0.00834 

9.85361 

3 

33 

42 
56 
7.0 

28 

9 

84540 

9 

99  191 

25 
26 
25 

0.00809 

985349 

32 

•3 

•4 

29 

30 

9 

84553 

13 
13 

9 

99217 

0.00  783 

9  85  337 

■3 

31 
30 

9 

84566 

9 

99242 

0.00  758 

9  85324 

31 

9 

84579 

*3 

9 

99267 

25 
26 

0.00  733 

9  85312 

29 

8  4 

32 

9 

84592 

9 

99293 
99318 

0.00  707 

9.85299 

28 

.7 

9.8 

33 

9 

84605 

13 

9 

25 

0.00  682 

9.85287 

27 

8 

II  .2 

34 

9 

84618 

13 
12 

9 

99  343 

25 

25 

26 

0.00657 

9.85274 

13 

t2 

26 

25 

.9 

12.6 

3S 

9 

84630 

9 

99368 

0.00632 

9 .  85  262 

36 

9 

84643 
84656 

13 

9 

99  394 

0.00606 

9.85250 

24 

1 

37 

9 

13 

9 

99419 

25 

0.00  581 

9  85237 

IJ 

23 

13 

38 

9 

84669 

13 

9 

99  444 

25 

0.00  556 

9. 85  225 

22 

I 

I  1 

39 
40 

9 

84682 

13 
12 

9 

99469 

25 

26 

0.00531 

9.85  212 

13 

t2 

21 
20 

.2 
.3 

3  9 

9 

84694 

9 

99  495 

0 .  00  505 

9.85  200 

41 

9 

84  707 

»3 

9 

99520 

25 

0 .  00  480 

9  85 187 

13 

19 

.4 

5.2 

42 

9 

84  720 

9 

99  545 

25 

0.00455 

9  85175 

18 

M 

43 

9 

84733 

13 

9 

99570 

25 

26 
25 

0.00430 

9  85 162 

13 

17 

.6 

44 
4S 

9 

84745 

13 

9 

99596 

0 .  00  404 

9  85 150 

'3 

16 
15 

•  7 
.8 

91 
10.4 

9 

84758 

9 

99621 

0.00379 

9  85 137 

46 

9 

84771 

»3 

9 

99646 

25 

0.00354 

9.85  125 

14 

9 

II. 7 

:i 

9 

84784 

13 

9 

99672 

0  00  328 

9.85  112 

13 

13 

1 

9 

84796 

9 

99697 

25 

0  00  303 

9.85 100 

12 

1 

49 

60 

9  84  809 

13 
13 

9 

99722 

25 

25 

0.00  278 
0.00  253 

9.85087 

13 
3 

II 
10 

la 

12 

9  84822 

9 

99  747 

9.85  074 

SI 

9 

84835 

13 

9 

99  773 

26 

0.00  227 

9 .  85  062 

9 

.2 

2.4 

S2 

9 

84847 

9 

99  798 

25 

0 .  00  202 

9,85049 

3 

8 

•3 

3^ 

S3 

9 

84860 

13 

9 

99823 

25 

0.00177 

9  85037 

7 

•4 

4.8 

54 
SS 

9 

84873 

13 
12 

9 

99848 

25 
26 

0.00  152 

9  85  024 

3 

2 

6 

5 

i 

6.0 

96 

10.8 

9 

84885 
84898 

9 

99874 

0.00  126 

9.85  012 

S6 

9 

13 

9 

99899 

25 

0.00  lOI 

984999   ; 

3 

4 

■7 
.8 

■>7 

9 

84  911 

13 

9 

99924 

25 

0.00076 

984986   ' 

3 

3 

S8 

9 

84923 

9 

99  949 

25 

0.00  051 

9.84974 

2 

•9 

59 
00 

9 

84936 

13 
13 

9 

99  975 

26 
25 

0.00  025 

9.84961    ] 

3 

2 

_0_ 

9 

84949 

0 

00  000 

0.00000 

9.84949 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tanjr. 

L.  Sin.      d 

[. 

Prop.  Pts. 

45° 

TABLE  III 


NATURAL 

TRIGONOMETRIC  FUNCTIONS 


FOR 


EACH    MINUTE 


71 


72 


TABLE  m 


©° 

1-- 

30 

30  ■  — 

4»     1 

t 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

60 

59 
58 

55 
54 

53 
52 
51 
50 

% 

47 
46 
45 
44 
43 
42 

41 
40 

35 
34 
33 
32 
31 
30 

27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

'\l 

15 
14 
13 
12 

0 

I 

2 

3 
4 

1 

9 

lO 

II 

12 
~^ 

14 

\l 

19 

20 
21 
22 
23 

26 

27 
28 
29 
30 

.00000 
.00029 
.ocx)58 
.00087 
.00116 
.00145 
.00175 

I. 00000 
I .00000 
I .00000 
I. 00000 
I .00000 
I. 00000 
I. 00000 

■01745 
.01774 
.01803 
.01832 
.01862 
.0189J 
.01920 

•99985 
.99984 
.99984 
.99983 

■99983 
.99982 
.99982 

•03490 
•03519 
■03548 
■03577 
.03606 

.03664 

•99939 
.99938 

■99937 
■99936 
■99935 
•99934 
•99933 

•05234 
.05263 
.05292 
■05321 
•05350 
•05379 
.05408 

.99863 
99861 
.99860 
.99858 
■99857 
■99855 
.99854 

.06976 
■07005 

■07034 
.07063 
.07092 
.07121 

.07150 

■99756 
■99754 
■99752 
■99750 
■99748 
■99746 
■99744 

.00204 
.00233 
.00262 
.00291 
.00320 
.00349 

I. 00000 
I. 00000 
I. 00000 
I. 00000 
.99999 
.99999 

.01949 
.01978 
.02007 
.02036 
.02065 
.02094 

.99981 
.99980 
.99980 
.99979 
99979 
■99978 

.03693 
.03723 
■03752 
■03781 
.03S10 

•03839 

.99932 

99931 
.99930 
.99929 
.99927 
.99926 

■05437 
.05466 

■05495 
■05524 
■05553 
•05582 

■99852 
.99851 
.99849 

•99847 
.99846 
.99844 

. .07208 

■07237 
.07266 
.07295 
•07324 

.99742 
.99740 
■99738 
.99736 

■99734 
•99731 

.00378 
.00407 
.00436 
.00465 
.00495 
.00524 

.99999 
.99999 
.99999 
.99999 

•99999 
.99999 

.02123 
.02152 
.02181 
.02211 
.02240 
.02269 

.99977 

•99977 
.99976 
.99976 
•99975 
■99974 

.03868 
.03897 
.03926 

•03955 
.03984 
.04013 

.99925 
.99924 
.99923 
.99922 
.99921 
.99919 

.05611 
.05640 
.05669 
.05698 
.05727 
■05756 

.99842 
.99841 
.99839 
.99838 
.99836 
.99834 

•07353 
.07382 
.07411 
.07440 
.07469 
.07498 

.99729 
.99727 
•99725 
•99723 
.99721 
.99719 

•00553 
.00582 
.00611 
.00640 
.00669 
.00698 

•99998 
.99998 
.99998 
•99998 
.99998 
.99998 

.02298 
.02327 
.02356 
.02385 
.02414 
.02443 

.99974 

■99973 
.99972 
.99972 
.99971 
.99970 

.04042 
.04071 
.04100 
.04129 
.04159 
.04188 

.99918 
.99917 
.99916 
■99915 
■99913 
.99912 

■05785 
.05814 
.05844 

■05873 
.05902 

■05931 

99833 
.99831 
.99829 
.99827 
.99826 
.99824 

■07527 
■07556 
■07585 
.07614 
.07643 
.07672 

.99716 
.99714 
.99712 
.99710 
.99708 
■99705 

.00727 
.00756 
.00785 
.00814 
.00844 
.00873 

.99997 
.99997 
.99997 

•99997 
.99996 
.99996 

.02472 
.02501 
.02530 
.02560 
.02589 
.02618 

.99969 

.99967 
.99966 
.99966 
.99965 
.99964 
.99963 
.99963 
.99962 
.99961 

.04217 
.04246 
.04275 
.04304 

•04333 
.04362 

.99911 
.99910 
.99909 
.99907 
.99906 
■99905 

.05960 
.05989 
.06018 
.06047 
.06076 
.06105 

.06163 
.06192 
.06221 
.06250 
.06279 

.99822 
.99821 
.99819 

.99817 
.99815 
.99813 

.07701 
•07730 
•07759 
.07788 
.07817 
.07846 

.99703 
.99701 
.99699 
.99696 
.99694 
.99692 

31 

32 

33 
34 

% 

39 
40 

41 

42 

43 
44 

% 

49 
50 
51 
52 
53 
54 

55 
56 

.00902 
.00931 
.00960 
.00989 
.01018 
.01047 

.99996 
.99996 
•99995 
•99995 
•99995 
•99995 

.02647 
.02676 
.02705 
•02734 
.02763 
.02792 

.04391 
.04420 
.04449 
.04478 

•04507 
•04536 

.99904 
.99902 
.99901 
.99900 
.99898 
.99897 

.99812 
.99810 
.99808 
.99806 
.99804 
.99803 

■07875 
.07904 

■07933 
.07962 
.07991 
.08020 

.99689 
■99687 
.99685 
.99683 
.99680 
.99678 

.01076 
.01105 

•01134 
.01164 
.01193 
.01222 

.99994 
.99994 

■99994 
•99993 
•99993 
■99993 

.02821 
.02850 
.02879 
.02908 
.02938 
.02967 

.99960 
■99959 
•99959 
.99958 

■99957 
■99956 

■04565 
■04594 
.04623 

•04653 
.04682 
.04711 

.99896 
.99894 
.99893 
.99892 
.99890 
.99889 

.06308 

^06366 

■06395 
.06424 

•06453 

.99801 
.99799 
.99797 
•99795 
•99793 
.99792 

.08049 
.08078 
.08107 
.08136 
.08165 
08194 

.99676 

■99673 
.99671 
.99668 
.99666 
.99664 
.99661 
•99659 
■99657 
.99654 
.99652 
.99649 

.01251 
.01280 
.01309 

•01338 
.01367 
.01396 

.99992 
.99992 
.99991 
.99991 
.99991 
.99990 

.02996 
.03025 

03054 
.03083 
.03112 
.03141 

•99955 
■99954 
■99953 
.99952 
.99952 
■99951 

.04740 
.04769 
.04798 
.04827 
.04856 
.04885 

.99888 
.99886 
■99885 
■99883 
.99882 
.99881 

.06482 
.06511 
.06540 
.06569 
.06598 
.06627 
.06656 
.06685 
.06714 
.06743 
■06773 
.06802 
.06831 
.06860 
.06889 
.06918 
.06947 
.06976 

.99790 
.99788 
•99786 
.99784 
.99782 
.99780 

■99778 
.99776 
■99774 
■99772 
■99770 
.99768 

.08223 
.08252 
.08281 
.08310 

■08339 
.08368 

.01425 
.01454 
.01483 

•01513 
.01542 
.01571 

.99989 
.99989 
.99989 

.03170 
.03199 
.03228 

.03286 
.03316 

•99950 

•99949 
.99948 

•99947 
.99946 

■99945 

.04914 

•04943 
.04972 
.05001 
.05030 
•05059 

.99879 
.99878 
.99876 
.99875 
■99873 
.99872 

.08397 
.08426 

•08455 
.08484 

•08513 
.08542 

.99647 

•99644 
.99642 

•99639 
•99637 
•99635 

II 
10 

I 

5 
4 

3 
2 

I 
0 

.01600 
.01629 
.01658 
.C1687 
.01716 
•01745 

.99987 
.99987 
.99986 
.99986 
99985 
■99985 

•Q3345 
•03374 
■03403 
•03432 
.03461 
.03490 

.99944 

•99943 
.99942 
.99941 
.99940 
99939 

.05088 
.05117 
.05146 

■05175 
.05205 

■05234 

.99870 
.99869 
.99867 
.99866 
.99864 
■99863 

.99766 
.99764 
.99762 
.99760 
•99758 
99756 

.08571 
.08600 
.08629 
.08658 
.08687 
.08716 

■99632 
■99630 
.99627 
.99625 
.99622 
.99619 

N.  cos. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

89° 

SS** 

87° 

86° 

85° 

TRIGONOMETRIC  FUNCTIONS  FOR  EACH  MINUTE 


73 


^^■^ 

5«     1 

6°    1 

7-    1 

8»    1 

»•      1 

60 

11 

55 
54 

53 

52 
51 
50 

It 
% 

45 
44 
43 
42 

o 

I 

2 

3 
4 

i 
I 

9 

lO 
i2 

13 

14 

\l 
\l 

19 

20 
21 
22 

23 
24 

1 

29 

30 

31 
32 

33 

34 

11 
11 

39 
40 

41 
42 

NT.  sine  ] 

ST.  COS. 

N.  sine  ] 

ST.  COS. 

NT.  sine] 

ST.  COS. 

^r.  sine|] 

ST.  COS. 

N.  sine 

N.  COS. 

.08716 

.08745 
.08774 
.08803 
.08831 
.08860 
.08889 

.99619 
.99617 
.99614 
.99612 
.99609 
.99607 
.99604 

.  10482 
.10511 

■  10540 

■  10569 

•10597 
.  10626 

•99452 
•99449 
•99446 
•99443 
•99440 
99437 
•99434 

.12187 
.12216 

.12245 
.12274 
.12302 

•12331 
.12360 

•99255 
.99251 
.99248 
.99244 
.99240 
.99237 
■99233 

•I39I7 
.13946 

•13975 

.14004 
•  14033 

.14061 

. 14090 

.99027 
.99023 
.99019 
.99015 
.99011 
.99006 
.99002 

•15643 
.15672 
.15701 
•15730 
.15758 
.15787 
.15816 

.98769 
.98764 
.98760 

•98755 
.98751 
.98746 
.98741 

.08918 
.08947 
.08976 
.09005 
.09034 
.09063 

.99602 

•99599 
.99596 

•99594 
.99591 
.99588 

.10713 
.10742 
.10771 
. 10800 

•99431 
.99428 

.99424 
.99421 
.99418 
■99415 

.12389 
.12418 
.12447 
.12476 
.12504 
•12533 

.99230 
.99226 
.99222 
.99219 
.99215 
.99211 
.99208 
.99204 
.99200 
.99197 

•99193 
.99189 

.14119 
.14148 

.14177 
•  14205 
.14234 
.14263 

.98998 
.98994 
.98990 
.98986 
.98982 
.98978 

.15845 

•15873 
.15902 

•15931 

•98737 
•98732 
.98728 
.98723 
.98718 
.98714 

.09092 
.09121 
.09150 
.09179 
.09208 
.09237 

.99586 

•99583 
.99580 

•99578 
•99575 
•99572 

. 10829 
'10887 

.10916 

.10945 
.10973 

.99412 

•99409 
.99406 
.99402 
•99399 
•99396 

.12562 
.12591 
.12620 
.12649 
.12678 
.12706 

. 14292 
.14320 
■14349 
•14378 
.14407 
.14436 

•98973 
.98969 
.98965 
.98961 
•98957 
•98953 

.16017 
.16046 
.16074 
.16103 
.16132 
.16160 

.98709 
.98704 
.98700 
.98695 

.09266 
.09295 
.09324 

•09353 
.09382 
.09411 

•99570 
•99567 
•99564 
.99562 

•99559 
•99556 

.11002 

.11031 

.11060 
. 1 1089 
.IIII8 
.11147 

•99393 
•99390 
.99386 

•99383 
.99380 

•99377 

■12735 
.12764 
.12793 
.12822 
.12851 
.12880 

.99186 
.99182 
.99178 

•99175 
.99171 
.99167 

.14464 

•14493 
.14522 

.14551 
.14580 
. 14608 

.98948 
.98944 
.98940 
.98936 
.98931 
.98927 

.16189 
.16218 
.16246 
.16275 
.16304 
•16333 

.98681 
.98676 
.98671 
.98667 
.98662 
•98657 

41 

40 

P 

.09440 
.09469 
.09498 
.09527 
•09556 
■09585 

■99553 
•99551 
.99548 

•99545 
.99542 
.99540 

. II 1 76 
.11205 
.11234 

.11263 

.11291 
.11320 

•99374 
•99370 
■99367 
.99364 
•99360 
•99357 

.12908 
.12937 
.12966 
.12995 
.13024 
•13053 

.99163 
.99160 
.99156 
.99152 
.99148 
.99144 

.14666 
.14695 
•14723 
•14752 
.14781 

■98923 
.98919 
.98914 
.98910 
.98906 
.98902 

.16361 
.16390 
.16419 
.16447 
.16476 
•16505 

.98652 
.98648 

•98643 
.98638 

•98633 
.98629 

35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 

23 
22 
21 
20 
19 

\l 

15 
14 
13 
12 

.09614 
.09642 
.09671 
.09700 
.09729 
.09758 

■99537 
•99534 
•99531 
.99528 
.99526 
•99523 

•I 1349 
.11378 
.11407 
.11436 
.11465 
.11494 

•99354 
•99351 
•99347 
•99344 
•99341 
•99337 

.13081 
.13110 

•13197 
.13226 

.99141 
•99137 
•99133 
.99.129 
.99125 
.99122 

.14810 

•  14838 

.14867 

.14896 
.14925 

•  14954 

.98897 

.98884 
.98880 
.98876 

•16533 
.16562 
.16591 
.16620 
.16648 
.16677 

.98624 
.98619 
.98614 
.98609 
.98604 
.98600 

.09787 
.09816 
.09845 
.09874 
.09903 
.09932 

.99520 
•99517 
•99514 
•9951 1 
.99508 
.99506 

•11523 

•II552 
.11580 

.11609 

.11638 
.11667 

•99334 
•99331 
.99327 

•99324 
.99320 

■99317 

•13254 
•13283 
•13312 
•13341 
•13370 
•13399 

.99118 
.99114 
.99110 
.99106 
.99102 
.99098 

.14982 
.15011 

.15040 
.15069 
•15097 

.15126 

.98871 
.98867 
.98863 
.98858 
.98854 
.98849 

.16706 
•16734 
.16763 
.16792 
.16820 
.16849 

•98595 
.98590 
.98585 
.98580 

•98575 
.98570 

43 
44 

11 

.09961 
.09990 
.10019 
.10048 
.10077 
.10106 

•99503 
•99500 
.99497 

•99494 
.99491 
.99488 

.11696 
.11725 

■11754 
.11783 
.11812 

.11840 

■99314 
.99310 

•99307 
•99303 
.99300 
.99297 

•13427 
•13456 
•13485 
•13514 
•13543 
•13572 

•99094 
.99091 
.99087 
•99083 
•99079 
•99075 

•I5I55 
.15184 

.15212 

.15241 
.15270 
•15299 

.98845 
.98841 
.98836 
.98832 
.98827 
.98823 

.16878 
.16906 

•16935 
.16964 
.16992 
.17021 

.98565 
.98561 
.98556 
.98551 
.98546 
.98541 

49 
50 
51 
52 
53 
54 

55 
56 

11 

i. 

.10135 
.10164 
.10192 
.10221 
.10250 
.10279 

•99485 
.99482 
.99479 
.99476 

•99473 
■99470 

.11869 
.11898 
.11927 
.11956 
.11985 

.12014 

•99293 
.99290 
99286 
.99283 
.99279 
.99276 

.13600 
.13629 

.13716 
•13744 

.99071 
.99067 
.99063 
.99059 

•99055 
.99051 

•15327 
•15356 

•15385 

•I54I4 
.15442 

•I547I 

.98818 
.98814 
.98809 
.98805 
.98800 
•98796 

.17050 
.17078 
.17107 
.17136 
.17164 
•17193 

•98536 
•98531 
.98526 
.98521 
.98516 
.98511 

10 

I 

5 
4 
3 
2 

0 

.10308 

•10395 
. 10424 
•10453 

■99467 
.99464 
.99461 
.99458 
■99455 
■99452 

.12043 
.12071 
.12100 
.12129 
.12158 
.12187 

.99272 
.99269 
•99265 
.99262 
.99258 
99255 

.13889 
•13917 

.99047 
.99043 
.99039 

•99035 
.99031 
.99027 

•15500 
•15529 

•'5557 
.15586 

•15615 
•15643 

.98791 

•98787 
.98782 
.98778 

•98773 
.98769 

. 1 7222 
.17250 
.17279 
.17308 
•17336 
•17365 

.98506 
.98501 
.98496 

.98481 

N.  COS. 

N.  sine 

N.  COS. 

M.  sine 

N.  COS. 

N.,sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

/ 

84'*    1 

§3°    1 

82°    1 

81°    1 

80° 

74 


TABLE  III 


t 

o 

I 

2 

3 
4 

I 
I 

9 

lO 

II 

12 

14 

\l 

17 

i8 

10°   j 

If 

19'        1 

1»» 

.4^ 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

!^.  sine 

N.  COS. 

N.  sine 

N.  COS. 

60 
59 
58 
57 

56 
55 

54 

53 
52 
51 
50 

It 

47 
46 

45 
44 
43 

42 

17365 
17393 
.17422 

17451 
■17479 
.17508 

•17537 

.98481 
.98476 
.98471 
.98466 
.98461 

98455 

•98450 

.19081 
.19109 
.19138 
.19167 

•19195 
. 19224 
.19252 

.98163 

.98157 
.98152 
.98146 
.98140 

98.35 

.98129 
.98124 
.98118 
.98112 
.98107 
.98101 
.98096 

20791 
.  20820 
.20848 
.20877 
.20905 
.20933 
.20962 

•97815 
.97809 

.97803 

.97797 
.97791 
•97784 
•97778 

•22495 
•22523 
•22552 
.22580 
.22608 
.22637 
.22665 

•97437 
•97430 
.97424 

•97417 
.97411 

•97404 
•97398 

.24192 
.24220 
.24249 
•24277 
•24305 
•24333 
.24362 

.97030 
•97023 

•97015 
.97008 
.97001 
■96994 
.96987 

■17565 
•17594 
.17623 
.17651 
.17680 
.17708 

.98445 
.98440 

•98435 
.98430 
.98425 

.98420 

.19281 
.19309 
•19338 
.19366 

•19395 
.19423 

.20990 
.21019 
.21047 
.21076 
.21104 
.21132 

.97772 
.97766 
.97760 

•97754 
.97748 
.97742 

.22693 

.22722 
.22750 
•22778 
.22807 
.22835 

•97391 
•97384 
•97378 
•97371 
•97365 
•97358 

•24390 
.24418 
.24446 
.24474 
•24503 
•24531 

.96980 

•96973 
.96966 

•96959 
.96952 
.96945 

.17766 

•17794 
.17823 
.17852 
.17880 

.98414 
.98409 
.98404 
•98399 

tut 

.19452 
.19481 
.19509 

.19566 
•19595 

.98090 
.98084 
.98079 
.98073 
.98067 
.98061 

.21161 
.21189 
.21218 
.21246 
.21275 
.21303 

■97735 
•97729 
•97723 
.97717 
.97711 
•97705 

.22863 
.22892 
.22920 
.22948 
.22977 
•23005 

•97351 
•97345 
•97338 
•97331 
•97325 
•97318 

•24559 
.24587 

•24615 
.24644 
.24672 
.24700 

•96937 
.96930 
.96923 
.96916 
.96909 
.96902 

19 

20 
21 
22 
23 

_^  - 
25 
26 
27 
28 
29 
30  . 

31 

32 

33 
34 

It 

.17909 

•17937 
.17966 

•17995 
.18023 
.18052 

•98383 
■98373 

:98368 
.98362 
•98357 

.19623 

.19680 
.19709 

.19766 

.98056 
.98050 
.98044 
•98039 
•98033 
.98027 

•21331 
.21360 
.21388 
.21417 
.21445 
.21474 

.97698 
.97692 
.97686 
.97680 

.97667 

•23033 
.23062 
.23090 
.23118 
.23146 
•23175 

973" 

•97304 
.97298 
.97291 
.97284 
•97278 

.24728 

.24756 
.24784 
.24813 
.24841 
.24869 

!  9688 7 
.96880 

•96873 
.96866 
.96858 

41 
40 

It 
11 

.18081 
.18109 
.18138 
.18166 
.18195 
.18224 

•98352 
•98347 
.98341 

•98336 
•98331 
•98325 

.19794 
.19823 

.19908 
•19937 

.98021 
.98016 
.98010 
.98004 
.97998 
.97992 

.21502 
•21530 
•21559 
.21587 
.21616 
.21644 

.97661 

.97648 
.97642 
.97636 
•97630 

.23203 

.23260 
.23288 
.23316 
•23345 

.97271 
.97264 
•97257 
•97251 
.97244 

•97237 

.24897 
.24925 
.24954 
.24982 
.25010 
.25038 

.96851 
.96844 

•96837 
.96829 
.96822 
.96815 

35 
34 
33 
32 
31 
30_ 

29 
28 

27 
26 
25 

24 

.18252 
.18281 

.18367 
•18395 

.98320 

•98315 
.98310 

•98304 
.98299 
.98294 

.19965 
.19994 
.20022 
.20051 
.20079 
.20108 

.97987 
.97981 

•97975 
.97969 

•97963 
•97958 

.21672 
.21701 
.21729 
.21758 
.21786 
.21814 

•97623 
.97617 
.97611 
.97604 
•97598 
•97592 

23373 
.23401 

•23429 
•23458 
•23486 
•23514 

.97230 
•97223 
.97217 
.97210 
•97203 
.97196 

.25066 
.25094 
.25122 
•25151 
•25179 
•25207 

.96807 
.96800 

till 
•96778 
.96771 

11 

39 
40 

41 
42 

.18424 
^18481 

•IP 

.18567 

.98288 
•98283 
.98277 
.98272 
.98267 
.98261 

.20136 
.20165 

•20193 
.20222 
.20250 
.20279 

•97952 
•97946 
.97940 

•97934 
.97928 
.97922 

•21843 
.21871 
.21899 
.21928 
.21956 
.21985 

•97585 
•97579 
•97573 
•97566 
.97560 

•97553 

•23542 
•23571 
•23599 
.23627 

.97189 
.97182 
.97176 
.97169 
.97162 
•97155 

•25235 
.25263 
.25291 
•25320 
•25348 
•25376 

.96764 
.967.56 

•96749 
.96742 

•96734 
.96727 

23 
22 
21 
20 
19 
18 

17 
16 

15 
H 
13 
12 

II 

10 

I 

5 
4 
3 
2 
I 
0 

43 
44 
45 

1 

49 
50 
51 
52 
53 
54 

11 

.18652 
.18681 

18710 

.18738 

.98256 
.98250 

•98245 
.98240 
.98234 
.98229 

.20307 
.20336 
20364 
•20393 
.20421 
.20450 

.97916 
.97910 

.97899 

•97893 
•97887 

.22013 
.22041 
.22070 
.22098 
.22126 
•22155 

•97547 
•97541 
•97534 
•97528 
•97521 
•97515 

.23712 
.23740 
.23769 

:2^^^5^ 
•23853 

.97148 
.97141 

■97134 
.97127 
.97120 
•97"3 

•25404 
•25432 
.25460 
.25488 
•25516 
•25545 

.96719 
.96712 
•96705 
.96697 
.96690 
.96682 

.18767 

'  18795 
. 18824 
.18852 
.18881 
.18910 

.98223 
.98218 
.98212 
.98207 
.98201 
.98196 

.20478 
.20507 

•20535 
.20563 
.20592 
.20620 

.97881 

•97875 
.97869 

•97863 
•97857 
■97851 

.22183 
.22212 

.22240 
.22268 
.22297 
•22325 

.97508 
.97502 
.97496 
.97489 

•97483 
.97476 

.23882 
.23910 

.23966 

•23995 
.24023 

.97106 
.97100 
.97093 
.97086 

•97079 
.97072 

•25573 
.25601 
.25629 

■.25685 
•25713 

.96675 
.96667 
.96660 

.96645 
.96638 

.18938 

.18967 

.18995 
.19024 
.19052 
.19081 

.98190 
.98185 
.98179 

tit 
.98163 

.20649 
.20677 
.20706 
.20734 
.20763 
.20791 

•97845 
•97839 
•97833 
•97827 
.97821 

97815 

•22353 
.22382 
.22410 
.22438 
.22467 
.22495 1 

.97470 
97463 
•97457 
.97450 
•97444 
97437 

.24051 
.24079 
.24108 
.24136 
.24164 
.24192 

.97065 
•97058 
•97051 
■97044 
•97037 
.97030 

•25741 
.25769 

'25826 

•25854 
.25882 

.96630 
.96623 

& 

96600 
■96593 

N.  COS. 

N.  sine 

M.  COS. 

N.  sine 

N.  cos.|N.  sine 

N.  COS. 

N.  sine 

N.  CCS. 

N.  sine 

/ 

1 

79-    1 

78^ 

yyo 

76° 

75° 

TRIGONOMETRIC  FUNCTIONS  FOR  EACH  MINUTE 


75 


o 
I 

2 

3 

4 

16°    j 

16"    1 

17° 

1§" 

19° 

60 

It 

11 

55 
54 

53 
52 
51 
50 

tt 
% 

45 
44 
43 
42 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

.25S82 
25910 

.25966 

•25994 
.26022 
.26050 

96593 
96585 
96578 
.96570 
.96562 
96555 
•96547 

.27564 
.27592 
.27620 
.27648 
.27676 
27704 
27731 

.96126 
.96118 
.96110 
.96102 

.96078 

29237 
.29265 
.29293 
.29321 
.29348 
.29376 
.29404 

•95630 
.95622 

•95613 
•95605 
•95596 
•95588 
•95579 

.30902 
.30929 
•30957 
•30985 
.31012 
.31040 
.31068 

.95106 

•95079 
.95070 
.95061 
.95052 

•32557 
•32584 
.32612 

.32694 
.32722 

•94552 
•94542 
•94533 
•94523 
•94514 
•94504 
•94495 

I 

9 

lO 

II 

12 

.26079 
.26107 
.26135 
.26163 
.26191 
.26219 

96540 
•96532 
.96524 

•96517 
.96509 
.96502 

■27759 
.27787 

•27815 
27843 
.27871 
.27899 

.96070 
.96062 
.96054 
.96046 
.96037 
.96029 

.29432 
.29460 
•29487 
•29515 
•29543 
■29571 

•95571 
•95562 
•95554 
■95545 
■95536 
•95528 

•31095 
•31123 

•31151 
.31178 
.31206 
•31233 

•95043 
•95033 
.95024 

•95015 
.95006 
.94997 

•32749 

.32777 
.32804 
•32832 

.94485 
.94476 
.94466 
•94457 
•94447 
•94438 

13 
14 

\i 

18 

19 
20 
21 
22 

23 

24 

.26247 
.26275 
.26303 
•26331 
•26359 
.26387 

•96494 
.96486 

.96479 
.96471 
.96463 
•96456 

.27927 

•27955 
.27983 
.28011 
.28039 
.28067 

.96021 
.96013 
.96005 

•95997 
.95989 
.95981 

.29599 
.29626 

.29710 
■29737 

•95519 
•95511 

•95502 

•95493 
•95485 
•95476 

.31261 
.31289 
•31316 
•31344 
•31372 
•31399 

.94988 

•94979 
.94970 
.94961 
.94952 
•94943 

.32914 
.32942 
•32969 
•32997 
•33024 
•33051 

.94428 
.94418 
.94409 

•94399 
.94390 
.94380 

•26415 
•26443 
.26471 
.26500 
.26528 
.26556 

.96448 
.Q6440 

•96433 
.96425 
.96417 
.96410 

.28095 
.28123 
.28150 
.28178 
.28206 
•28234 

■95972 
.95964 

•95956 
.95948 
•95940 
■95931 

.29765 

•29793 
.29821 
.29849 
.29876 
.29904 

•95467 
•95459 
•95450 
•95441 
■95433 
•95424 

•31427 
•31454 
.31482 
•31510 
•31537 
•31565 

•94933 
•94924 
•94915 
.94906 

'.lists 

•33079 
.33106 

•33134 
•33161 
•33189 
.33216 

•94370 
.94361 

•94351 
.94342 

•94332 
.94322 

41 

40 

39 

1 

25 

26 

li 

29 
30 

3J 
32 

33 
34 
35 
36 

11 

39 
40 

41 

43 
44 
45 
46 

47 
48 

.26584 
.26612 

^26696 
.26724 
.26752 
.26780 
.26808 
.26836 
.26864 
.26892 

.96402 

•96394 
.96386 

96379 
•96371 
96363 
•96355 
•96347 
.96340 
.96332 

•96324 
.96316 

.28262 
.28290 
.28318 
.28346 

•28374 
.28402 

■95923 
•95915 
•95907 
.95898 
.95890 
.95882 

■29932 
.29960 

•29987 
■30015 
■30043 
.30071 

.30098 
.30126 

■30154 
.30182 
.30209 
.30237 

•95415 

•95389 
•95380 
•95372 

•31593 
.31620 
.31648 
•31675 
•31703 
•31730 

.94860 
.94851 
.94842 
.94832 

•33244 
•33271 
•33298 
•33326 
•33353 
•33381 

•94313 
•94303 
•94293 
.94284 

•94274 
.94264 

•94254 
•94245 
•94235 
.94225 
.94215 
.94206 

35 
34 
33 
32 
31 
30 

27 
26 

25 
24 

23 
22 
21 
20 

\l 
15 
14 
13 
12 

II 

10 

\ 

5 
4 
3 
2 

I 
0 

.28429 

•28457 
.28485 
.28513 

.28569 

•95874 
•95865 
•95857 
•95849 
.95841 
■95832 

•95363 
■95354 
■95345 
95337 
■95328 
95319 

•31758 
.31786 
.31813 
.31841 
.31868 
•31896 

•94823 
.94814 
•94805 

•94795 
.94786 

■94777 

•33408 
•33436 
•33463 
•33490 
•33518 
•33545 

.26920 
.26948 
.26976 
.27004 
.27032 
.27060 

.96308 
•96301 
.96293 
.96285 
.96277 
.96269 

.28597 
.28625 
.28652 
.28680 
.28708 
•28736 

•95816 
•95807 
•95799 
•95791 
.95782 

.30265 
.30292 
■30320 
•30348 
.30376 
■30403 

95310 

•95301 

•95293 
.95284 

•95275 
.95266 

•31923 
•31951 
•31979 
.32006 

•32034 
.32061 

.94768 

•94758 
.94749 
.94740 

•94730 
.94721 

•33573 
.33600 
.33627 

•33710 

.94196 
.94186 
.94176 
.94167 

•94157 
.94147 

.27088 
.27116 
.27144 
.27172 
.27200 
.27228 

.96261 

•96253 
.96246 
.96238 
.96230 
.96222 

.28764 
.28792 
28820 
.28847 
.28875 
.28903 

.95766 
•95757 
•95749 
•95740 
■95732 

■30431 
■30459 
.30486 

■30514 
•30542 
•30570 

■95257 
.95248 
.95240 

■95231 
■95222 

95213 

.32089 
.32116 
.32144 
.32171 
.32199 
.32227 

.94712 
.94702 

•94693 
.94684 
.94674 
•94665 

•33737 
•33764 
•33792 
•33819 
•33846 
•33874 

•94137 
.94127 
.94118 
.94108 

49 
50 
51 

11 

54- 

11 

11 

.27256 
.27284 
.27312 
.27340 
.27368 
.27396 

.96214 
.96206 
.96198 
.96190 
.96182 
.96174 

•28931 

.29015 
.29042 
.29070 

•95724 
•95715 
•95707 
95698 
.95690 
.95681 

•30597 
.30625 

.30708 
•30736 

■95204 

•^5195 
.95186 

.95168 
•95159 

•32254 
.32282 
.32309 
•32337 
•32364 
.32392 

•94656 
.94646 

•94637 
.94627 
.94618 
.94609 

•33901 
•33929 
•33956 
•33983 
.54011 
■34038 

.94078 
.94068 
•94058 
•94049 
•94039 
.94029 

.27424 
.27452 
.27480 
.27508 

•27536 
.27564 

.96166 
.96158 
.96150 
.96142 

.96126 

.29098 
.29126 
.29154 
29182 
.29209 
.29237 

•95664 
•95656 
•95647 
95639 
•95630 

.30763 
.30791 
.30819 
.30846 
.30874 
.30902 

■95150 
■95142 
■95133 
■95124 

95"5 
.95106 

.32419 

•32447 
•32474 
.32502 

•32529 
•32557 

•94599 
•94590 
.94580 

•94571 

.94561 

•94552 

-34065 

•34093 
.34120 

•34147 
•34175 
.34202 

•94019 
.94009 

•93999 
•93989 
•93979 
.93969 

N.  COS. 

N.  sine 

N.  COS. 

N  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

/ 

74"    1    73°    1    72" 

71"    1    70" 

76 


TABLE  III 


f 

o 

I 

2 

3 
4 

"I 

9 

lO 

II 

12 

20°    1 

«» 

22°    1 

23°     1 

24° 

60 
59 
58 

11 

55 

il_ 

53 
52 
51 
50 
49 
48 

47 
46 

45 
44 
43 
42 

41 
40 
39 
38 

11 

35 
34 
33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

\t 
\l 

15 
14 
13 
12 

10 

I 

5 
4 
3 

2 
I 
0 

N.  sine 

^T.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

.34202 
.34229 

.34284 
•343" 
•34339 
•34366 

.93969 
•93959 
93949 
■93939 
.93929 

•93919 
•93909 

•35891 
■35918 
•35945 
•35973 
.36000 

•93358 
•93348 
•93337 
•93327 
•93316 
•93306 
•93295 

•37461 
.37488 

•37515 
•37542 
•37569 
•37595 
.37622 

.92718 

.92707 
.92697 
.92686 

.92664 
•92653 

•39073 
.39100 

•39127 

•39153 
.39180 
.39207 
■39234 

.92050 
.92039 
92028 
.92016 
.92005 
.91994 
.91982 

.40674 
.40700 
.40727 

•40753 
.40780 
.40806 
•40833 
.40860 
.40886 
.40913 

.40992 

•91355 
91343 
91331 
•91319 
•91307 
.9 [295 
.91283 

•34393 
.34421 
.34448 
•34475 
•34503 
•34530 

•93899 
.93889 

•93879 
.93869 

•93859 
•93849 

.36027 
.36108 

■93285 
•93274 
•93264 
•93253 
•93243 
.93232 

•37649 
.37676 

■37703 
•37730 
•37757 
•37784 

.92642 
.92631 
.92620 
.92609 
.92598 
•92587 

.39260 
.39287 
•39314 
•39341 
•39367 
■39394 

.91971 

•91959 
.91948 
.91936 
.91925 
.91914 

.91272 
.91260 
.91248 
.91236 
.91224 
.91212 

13 

14 

•34557 
•345^4 
.34612 

•34639 
.34666 
.34694 

•93839 
•93829 
.93819 
•93809 
•93799 
■93789 

.36190 
.36217 

•36244 
.36271 
.36298 
•36325 

.93222 
.93211 
•93201 

.93180 
.93169 

.37811 
•37838 
•37865 
.37892 

•37919 
•37946 

.92576 
.92565 
•92554 
•92543 
•92532 
.92521 

.39421 
.39448 
•39474 
•39501 
•39528 
•39555 

.91902 
.91891 
.91879 
.91868 
.91856 
.91845 

.41019 
.41045 
.41072 
.41098 
.41125 
.41151 

.91200 
.91188 
.91176 
.91164 
.91152 
.91140 

19 

20 
21 
22 

23 

24 

•34721 
•34748 
•34775 
•34803 
•34830 
•34857 

•93779 
•93769 
•93759 
•93748 
•93738 
.93728 

•36352 
•36379 
.36406 

.36461 
.36488 

•93159 
.93148 

•93137 
•93127 
.93116 
.93106 

•37973 

.38107 

.92510 

•92499 

.92488 
.92477 
.92466 
•92455 

•39581 
.39608 

.39688 
•39715 

•91833 
.91822 
.91810 
.91799 
.91787 
•91775 

.41178 
.41204 
.41231 
•41257 
.41284 
.41310 

.91128 
.91116 
.91104 
.91092 
.91080 
.91068 

29 

30 

31 

32 

33 
34 

P 

39 
40 

41 
42 

43 
44 
45 
46 

ti 

49 
50 
51 
52 
53 
54 

55 
56 

,34884 
.34912 

•34939 
.34966 

•34993 
•35021 

•93718 
.93708 
.93698 
.93688 
.93677 
.93667 

•36515 
•36542 
.36569 
.36596 

•^^?^ 
.36650 

•93095 
.93084 

•93074 
.93063 

•93052 
.93042 

.38188 
.38215 
.38241 
.38268 

•92444 
.92432 
.92421 
.92410 

•39741 
•39768 

•39795 
.39822 

•39848 

•39875 

.91764 

•91752 
.91741 
.91729 
.91718 
.91706 

•41337 
•41363 
.41390 
.41416 

•41443 
.41469 

.91056 
.91044 
.91032 
.91020 
.91008 
.90996 

•3^048 

•35075 
•35102 

•35130 
•35157 
•35184 

•93657 
•93647 
•93637 
.93626 
.93616 
.93606 

•36677 
•36704 
•36731 
•36758 

•93031 
.93020 
.93010 

.92978 

•38295 
.38322 

•38349 
•38376 
•38403 
•38430 

■$t 

.38510 
•38537 
.38564 
•38591 

■92377 
.92366 

•92355 
•92343 
.92332 
.92321 
.92310 
.92299 
.92287 
.92276 
.92265 
•92254 

•39902 
•39928 

•39955 
.39982 
.40008 
•40035 

.91694 
.91683 
.91671 
.91660 
.91648 
.91636 

.41496 
.41522 
•41549 
•41575 
.41602 
.41628 

.90984 
.90972 
.90960 
.90948 
.90936 
.90924 

•35211 
•35239 
.35266 

•35293 
•35320 
•35347 

•93596 
•93585 
•93575 
•93565 
•93555 
•93544 

•36894 
.36921 
.36948 
•36975 

.92967 
.92956 
•92945 
•92935 
.92924 
.92913 

.40062 
.40088 
.40115 
.40141 
.40168 
.40195 

.91625 
.91613 
.91601 
.91590 

.91566 

tell 

.41707 

•41734 
.41760 
.41787 

.90911 
.90899 
.90887 

•90875 
.90863 
.90851 

•35375 
•35402 

•35429 
•35456 
•35484 
•355" 

•93534 
•93524 
•93514 
•93503 
•93493 
•93483 

.37002 
.37029 
•37056 
•37083 
.37110 

•37137 

.92902 
.92892 
.92881 
.92870 
•92859 
.92849 

.38617 
.38644 
.38671 
.38698 
•38725 
•38752 

.92243 
.92231 
.92220 
.92209 
.92198 
.92186 

.40221 
.40248 
.40275 
.40301 
.40328 
•40355 

•91555 
•91543 
91531 
91519 
.91508 
.91496 

.41813 
.41840 
.41866 
.41892 
.41919 
•41945 

•90839 
.90826 
.90814 
.90802 
.90790 
■90778 

•35538 
•35565 
•35592 
•35619 
•35647 
•35674 

•93472 
•93462 
•93452 
•93441 
•93431 
.93420 

•37164 
•37191 
.37218 

•37245 
.37272 

•37299 

.92838 
.92827 
.92816 
•92805 
.92794 
.92784 

.38912 

•92175 
.92164 
.92152 
.92141 
.92130 
.92119 

.40381 
.40408 
.40434 
.40461 
.40488 
.40514 

.91484 
.91472 
.91461 
.91449 

•91437 
.91425 

.41972 
.41998 
.42024 
•42051 
•42077 
.42104 

.90766 

•90753 
.90741 

9^729 
.90717 
.90704 
.90692 
.90680 
.90668 
90655 
90643 
.90631 

•35701 
•35728 

•35755 
•35782 
•35810 
•35837 

.93410 
•93400 
•93389 
•93379 
•93368 
•93358 

•37326 
•37353 
•37380 
•37407 
•37434 
•37461 

.92773 
.92762 
.92751 
.92740 
.92729 
.92718 

•38939 
.38966 

•38993 
.39020 
.39046 
•39073 

.92107 
.92096 
.92085 
.92073 
.92062 
.92050 

.40541 
.40567 
.40594 
.40621 
.40647 
.40674 

.91414 
.91402 
.91390 

.91366 
•91355 

•42130 
.42156 
.42183 
.42209 

•42235 
.42262 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

/ 

69°    1 

68° 

67°    1 

66°    1 

65° 

TRIGONOMETRIC  FUNCTIONS  FOR  EACH  MINUTE 


77 


9 

35°    1 

26°    1 

sr      1 

28°    1 

29°    1 

60 
59 
58 
57 

56 
55 

54 

53 
52 
51 
50 

:i 

47 
46 

45 
44 
43 
42 

41 
40 

35 
34 
33 
32 
31 
30 

29 
28 

V, 

25 
24 

23 
22 
21 

20 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

o 
I 

2 

3 
4 
c 

l_ 
9 

lO 
12 

13 

14 

:i 

17 
18 

.42262 
.42288 
•42315 
•42341 
•42367 
.42394 
.42420 

.90631 
.90618 
.90606 
.90594 
.90582 
.90569 
•90557 

•43863 
■43889 
.43916 
■43942 
.43968 
•43994 

.89879 
.89867 

.89854 
.89841 
.89828 
.89816 
.89803 

45399 
•45425 
•45451 
■45477 
•45503 
•45529 
■45554 

.89101 
.89087 
.89074 
.89061 
.89048 

■89035 
.89021 

.46947 
•46973 
•46999 
.47024 
.47050 
.47076 
.47101 

.88295 
.88281 
.88267 
.88254 
.88240 
.88226 
.88213 

.48481 
.48506 
■48532 
■48557 
■48583 
.48608 
.48634 

■48659 
.48684 
.48710 

■48735 
.48761 
.48786 

.87462 
.87448 

■87434 
.87420 
.87406 
■87391 
■87377 
•87363 
•87349 
•87335 
.87321 
•87306 
.87292 

.42446 
•42473 
•42499 
•42525 
•42552 
.42578 

•90545 
•90532 
.90520 
.90507 
.90495 
.90483 

.44020 
.44046 
.44072 
.44098 
.44124 
•44151 
•44177 
■44203 
.44229 

■44255 
.44281 

•44307 

.89790 
•89777 
.89764 
•89752 
■89739 
.89726 

.45580 
.45606 
•45632 
■45658 
■45684 
•45710 

.89008 
.88995 
.88981 
.88968 

.88955 
.88942 

.47127 

•47153 
.47178 
.47204 
.47229 
•47255 

.88199 
.88185 
.88172 
.88158 

.88144 
.88130 

.42604 
•42631 

.42709 
•42736 

.90470 
.90458 
.90446 

•90433 
.90421 
.90408 

■89713 
.89700 

.89687 
.89674 
.89662 
.89649 

•45736 
•45762 
•45787 
•45813 
•45839 
•45865 

.88928 
•88915 

:  88888 
.88875 
.88862 

.47281 
.47306 
•47332 
•47358 
■47383 
■47409 

.88ii7 
.88103 
.88089 
.88075 
.88062 
.88048 

.48811 

•48837 
.48862 
.48888 
•48913 
•48938 

.87278 
.87264 
.87250 

•87235 
.87221 
.87207 

19 
20 
21 
22 

23 

24 

.42762 
.42788 

•42815 
.42841 
.42867 
•42894 

•90396 
•90383 
•90371 
•90358 
.90346 

•90334 

•44333 
•44359 
•44385 
.44411 

•44437 
•44464 

.89636 
.89623 
.89610 

•89597 
.89584 
.89571 

.45891 
•45917 
•45942 
.45968 
•45994 
.46020 

.88848 
•88835 
.88822 
.88808 

:i5i 

■47434 
.47460 
■47486 
■475" 
■47537 
■47562 
•47588 
.47614 

•47639 
.47665 
.47690 
.47716 

.88034 
.88020 
.88006 

■87993 
.87979 

.87965 

.48964 
.48989 
.49014 
.49040 
•49065 
.49090 

•87193 
.87178 
.87164 
.87150 
.87136 
.87121 

25 
26 

27 
28 
29 
3c 

31 

32 
33 
34 

% 

39 
40 

41 
42 

43 
44 

47 
48 

49 
50 
51 
52 
53 
54 

.42920 
.42946 
.42972 
.42999 
•43025 
•43051 

.90321 
.90309 
.90296 
.90284 
.90271 
•90259 

■44490 
.44516 
■44542 
•44568 
•44594 
.44620 

.89558 
■89545 
■89532 
.89519 
.89506 
.89493 

.46046 
.46072 
.46097 
.46123 
.46149 
•46175 

.88768 
•88755 
.88741 
.88728 

.88715 
.88701 

•87951 
•87937 

li 

".87882 

.49116 
.49141 
.49166 
.49192 
.49217 
.49242 

.87107 
.87093 
.87079 
.87064 
.87050 
•87036 

•43077 
.43104 

•43130 
•43156 
.43182 
.43209 

.90245 
.90233 
.90221 
.90208 
.90196 
.90183 

.44646 
•44672 
.44698 
.44724 
•44750 
.44776 

.89480 
.89467 

■89454 
.89441 
.89428 
.89415 

.46201 
.46226 
.46252 
.46278 
•46304 
•46330 

.88688 
.88674 
.88661 
.88647 
.88634 
.88620 

•47741 
•47767 

.47844 
•47869 

•47895 
.47920 
.47946 
•47971 
•47997 
.48022 

.87868 

•87854 
.87840 
.87826 
.87812 
•87798 

•87784 
.87770 

.87756 

■87743 
.87729 

.87715 

.49268 
.49293 
.49318 
•49344 
•49369 
•49394 

.87021 
.87007 

•86993 
.86978 
.86964 
.86949 

•43235 
.43261 
.43287 

•43313 
■43340 
•43366 

.90171 
.90158 
.90146 

•90133 
.90120 
.90108 

.44802 
.44828 
.44854 
.44880 
.44906 
.44932 

■44958 
.44984 
45010 
•45036 
■45062 
.45088 

.89402 
.89389 
■89376 
•89363 
•89350 
•89337 
.89324 
.89311 
.89298 
.89285 
.89272 
.89259 

•46355 
.46381 
.46407 

•46433 
.46458 
.46484 

.88607 

.88566 
.88539 

.49419 

•49445 
•49470 

•49495 
.49521 

■49546 

•86935 
.86921 

.86878 
.86863 

•43392 
.43418 

•43445 
•43471 
•43497 
•43523 

.90095 
.90082 
.90070 
.90057 
.90045 
.90032 

.46510 
•46536 
•46561 
.46587 
.46613 
■46639 

.88526 
.88512 

.88499 
.88485 
.88472 
.88458 

.48048 

•48073 
.48099 
.48124 
.48150 
•48175 

.87701 
.87687 
■87673 
■87659 
•^645 
.87631 

■49571 
■49596 
.49622 
.49647 
.49672 
.49697 

■49723 
■49748 

•49773 
.49798 
.49824 
•49849 

.86849 
.86834 
.8682c 
.86805 
.86791 
.86777 

15 
14 
13 
12 

11 

10 

7 
6 

•43549 
•43575 
.43602 
•43628 

.90019 
.90007 

:& 

.89968 
.89956 

•89943 
•89930 
.89918 

■89905 
.89892 

•89879 

■45114 
.45140 
.45166 
.45192 
.45218 
■45243 

■89245 
■89232 
.89219 
.89206 

.89193 
.89180 

.46664 
.46690 
.46716 
.46742 
.46767 
■46793 

.88445 
.88431 
.88417 
.88404 
.88390 
•88377 

.48201 
.48226 
.48252 
.48277 

■48303 
.48328 

.87617 
.87603 
.87589 
•87575 
.87561 
.87546 

.86762 
.86748 
■86733 
.86719 
.86704 
.86690 

•43706 
43733 
43759 
.43785 
.43811 

•43837 

•45269 
•45295 
•45321 
•45347 
•45373 
•4539S 

.89167 

■89153 
.89140 
.89127 
.89114 
.89101 

.46819 
.46844 
.4687c 
.46896 
.46921 
■46947 

.88363 

.88349 
■88336 
.88322 
.88308 
.88295 

•48354 
•48379 
.48405 
.4843c 
■48456 
.48481 

•87532 
.87518 
.87504 
.87490 
•87476 
.87462 

.49874 
.49899 
.49924 
.49950 

•49975 
.5000c 

.86675 
.86661 
.86646 
.86632 
.86617 
.86603 

5 
4 
3 

2 
I 
0 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

/ 

fij^ 

«;i° 

62° 

61° 

60° 

^^__ 

78 

TABLE  III 

SO''   1 

31°    1 

92^        1 

33° 

34° 

o 

I 

2 

3 
4 

I 
I 

9 

lO 

II 

12 

13 
14 

i6 

N.  sine  ] 

ST.  COS. 

N.  sine  ] 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

NT.  COS. 

N.  sine 

N.  COS. 

60 

^^ 

55 
54 

53 
52 
51 
50 

4I 

47 
46 

45 
44 
43 
42 

50000 
•5^5025 
.50050 
.50076 
50101 
50126 
50151 

.86603 
.86588 

8^573 
86559 
86544 
86530 
.86515 

•51504 
•51529 
•51554 
•5'579 

•5'653 

•85717 
.85702 
.85687 
.85672 
85657 
•85642 
.85627 
.85612 

•85597 
.85582 
.85567 
.85551 
•85536 

52992 
•530*7 

•53091 
53115 
•53140 

.84805 
•84789 
.84774 
.84759 

•84743 
.84728 
.84712 

•54513 
•54537 
.54561 
.54586 
.54610 

.83867 
•83851 
•83835 
.83819 

■83772 

•55919 

:^ 

•55992 
.56016 
.56040 
.56064 

.82871 
.82855 
.82839 
.82822 
.82806 

.50176 
.50201 
.50227 
.50252 
.50277 
.50302 

.86501 
.86486 
.86471 

.86457 
.86442 
.86427 

.51678 

•51703 
.51728 

5'753 
.51778 
.51803 

•53164 
53189 

.84697 
.84681 
.84666 
.84650 

•84635 
.84619 

■54635 

.54708 
54732 
■54756 

83756 
.83740 

83724 
.83708 
.83692 
■83676 

.56088 
.56112 

itt 

.56184 
.56208 

.82790 
82773 
•82757 
.82741 

mil 

50327 
50352 
•50377 
•50403 
.50428 

•50453 

86413 
.86398 
•86384 
.86369 
•86354 
.86340 

.51828 

.51852 

•51877 
.51902 
,51927 
•51952 

.85521 
.85506 
.85491 
.85476 
.85461 
.85446 

•53312 
•53337 
•53361 
•53386 

•534" 
•53435 

.84604 
.84588 
•84573 
•84557 
.84542 
.84526 

.54781 
.54805 
•54829 
•54854 
■54878 
■54902 

.83660 

•83645 
.83629 

•83613 
•83597 
.83581 

tit 
•56305 
•56329 
•56353 

.82692 
.82675 
.82659 
.82643 
.82626 
.82610 

19 

20 
21 
22 

23 

24 

^1 

27 
28 
29 

30 

.50478 

•50503 
.50528 

50553 
50578 
.50603 

.86325 
.86310 
.86295 
.86281 
.86266 
.86251 

51977 
.52002 
.52026 

52051 
.52076 
52101 

•85431 
.85416 
•85401 
■85385 
•85370 
•85355 

•53460 

•53484 

•53509 

•53534 

•53558 

•53583. 

•53607 

•53705 
•53730 

.84511 

•84495 
.84480 
.84464 
.84448 
•84433 

■54927 
•54951 
•54975 
■54999 
■55024 
■55048 

•83565 
•83549 
•83533 
•83517 
•83501 
.83485 

•56377 
.56401 
.56425 
•56449 
•56473 
•56497 

•82593 
.82577 
.82561 
•82544 
.82528 
.82511 

41 
40 

'^ 
1? 

35 
34 
33 
32 
31 
30 

29 
28 

V, 

25 
24 

23 
22 
21 
20 

18 

17 
16 

15 
14 
13 
12 

II 

10 

i 
I 

5 
4 
3 

2 
I 
0 

.50628 
.50654 
50679 
•50704 
.50729 

•50754 

86237 
.86222 
.86207 
.86192 
.86178 
.86163 

.52126 
52151 
•52175 
.52200 
.52225 
•52250 

•85340 

•85325 
.85310 
.85294 
.85279 
.85264 

.84417 
.84402 
.84386 
•84370 
•84355 
•84339 

■55072 
■55097 
.55121 

•55145 
•55169 
•55194 

•83469 
•83453 
•83437 
.83421 

•83405 
•83389 

•56521 
.56641 

•82495 
.82478 
.82462 
.82446 
.82429 
.82413 

3 

32 

33 
34 

11 

•50779 
.50804 
.  50829 
.50854 
.50879 
•50904 

.86148 
•86133 
.86119 
.86104 
.86089 
.86074 

•52275 
•52299 
•52324 
52349 
•52374 
•52399 

•85249 
.85234 
.85218 
.85203 
.85188 
•85173 

•53754 

•53828 
•53853 
•53877 

.84292 

•84277 
.84261 
.84245 

•55218 

.55266 
•55291 
55315 
•55339 

•83373 
•83356 
•83340 
•83324 
.83308 
.83292 

.56665 
.56689 

•56713 

.56760 
.56784 

.82363 
.82347 
.82330 
•82314 

11 

39 
40 

41 
42 

.50929 
■50954 
•50979 
.51004 
.51029 
•51054 

.86059 
.86045 
.86030 
86015 
.86000 
.85985 

•52448 
•52473 
•52498 
•52522 
•52547 

.85157 
.85142 
.85127 
.85112 
.85096 
.85081 

•53902 
•53926 
•53951 

•53975 
.54000 

•54024 

.84230 
.84214 
.84198 
.84182 
.84167 
.84151 

•55412 
•55436 
•55460 
•55484 

•83276 
.83260 

•83244 
.83228 
.83212 
•83195 

.56808 

•56832 

■IS 

.82297 
.82281 
.82264 
.82248 
.82231 
.82214 

43 
44 

tl 

47 
48 

49 
50 
51 
52 
53 
54 

55 
56 

•51079 
.51104 
•51129 
•51154 
•51179 
.51204 

.85970 
.85956 
■85941 
.85926 
.85911 
.85896 

•52572 
•52597 
52621 
.52646 
.52671 
.52696 

.85066 
.85051 

•85035 
.85020 
•85005 
.84989 

•54049 
•54073 
•54097 
•54122 
.54146 
•54171 

•84135 
.84120 
.84104 
.84088 
.84072 
•84057 

•55509 
•55533 
•55557 
.55581 

•55605 
•55630 

•83179 
•83163 
•83147 
•83J31 

;  83098 

.57000 
.57024 
•57047 
•57071 

.82198 
.82181 
.82165 
.82148 
.82132 
•82115 

.51229 
•51254 
•51279 
•51304 
•51329 
•51354 

•51379 
.51404 
.51429 
•5H54 
•5H79 
•51504 

.85881 
.85866 
.85851 
•85836 
.85821 
.85806 
.85792 

•85777 
.85762 

•85747 
•85732 
.85717 

.52720 

•52745 
.52770 

•52794 
.52819 
.52844 

•84974 
.84959 
•84943 
.84928 
.84913 
.84897 

•54195 
.54220 

•54244 
.54269 

•54293 
•54317 

.84041 
.84025 
.84009 
•83994 
•83978 
.83962 

•55654 
•55678 
•55702 
•55726 
•55750 
•55775 

.83082 
.83066 
•83050 
•83034 
•83017 
.83001 

•57095 
•57119 
•57143 
•57167 
•57I9I 
•57215 

.82098 
.82082 
.82065 
.82048 
.82032 
.82015 

.52869 

•52893 
.52918 

•52943 
.52967 
.52992 

.84882 
.84866 
.84851 
.84836 
.84820 
.84805 

•54366 
■54391 
•54415 
•54440 
■54464 

•83946 
•83930 

i%l 
.83883 
.83867 

•55799 
.55823 
•55847 
.55871 
•55895 
•55919 

.82985 
.82969 

•82953 
.82936 
.82920 
.82904 

•57238 
.57262 
.57286 
•57310 

•57334 
•57358 

.81999 
.81982 
.81965 
.81949 
.81932 
.81915 

N  COS. 

N.  sine 

N.  COS. 

Kfrgirie 

N.  COS. 

N.  sine 

N.  COS. 

M.  sine 

N.  COS. 

N.  sine 

p 

59"    1 

68**    1 

.«»     1 

56° 

55' 

TRIGONOMETRIC  FUNCTIONS  FOR  EACH  MINUTE 


79 


ss*'      1 

36° 

37°    1 

38°    1 

39° 

9 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  cos. 

O 

I 
2 

3 
4 

i 

7 
8 

9 

lO 
12 

13 

14 
15 
16 

57358 
•57381 
■57405 

57429 
•57453 
•57477 
■57501 

•81882 
.81865 
•81848 
.81832 
•81815 

•58779 
•58802 
.58826 
.58849 

.■|896 
.58920 

.80867 
.80850 
.80833 
.80816 
.80799 

60182 
. 60205 
.60228 
.60251 
•60274 
.60298 
.60321 

•79864 
•79846 
.79829 
•79811 

•79793 
•79776 

•79758 

.61566 

.61612 
61635 
.61658 
.61681 
.61704 

.78801 
•78783 
.78765 
•78747 
•78729 
•78711 
•78694 

.62932 

•62955 
.62977 
•63000 
•63022 

•63045 
•63068 

•77715 
. 77606 

60 
59 

.77678 
.77660 
.77641 
•77623 
•77605 

58 

55 
54 

53 
52 
51 
50 
49 
48 

57524 
•57548 
•57572 
•57596 
•57619 
•57643 

•81798 
•81782 
•81765 
•81748 

•81731 
•81714 

■58943 
.58967 

■58990 
.59014 

■59037 
.59061 

.80782 
.80765 
.80748 
•80730 

•80713 
.80696 

.60344 
.60367 
.60390 
.60414 
.60437 
.60460 

•79741 
•79723 
•79706 
•79688 
•79671 
•79653 

.61726 

•61749 
.61772 

•61795 
.61818 
.61841 

•78676 
•78658 
•  78640 
.78622 
.78604 
•78586 

•63090 
•63113 
•63135 
•63158 
.63180 
.63203 

•77586 
•77568 
•77550 
•77531 
•77513 
•  77494 

•57667 
•57691 
•57715 

•57762 
.57786 

•81698 
.81681 
•81664 
•81647 
•81631 
•81614 

.59084 
.59108 
■59131 
•59154 
•59178 
•59201 

.80679 
•80662 
.80644 
80627 
.80610 
•80593 

•60529 

•60576 
•60599 

•79635 
.79618 
•79600 
•79583 
•79565 
•79547 

.61864 
.61887 
•61909 
•61932 

•61955 
.61978 

•  78568 
•78550 
■78532 
•78514 
•78496 
•78478 

.63225 
•63248 
.63271 

IP 

•63338 

•77476 
•77458 
•77439 
•77421 
.77402 
•77384 

45 
44 
43 
42 

41 
40 

% 

35 
34 
33 
32 
31 
30 

19 
20 
21 
22 

23 
24 

^i 

27 
28 
29 
30 

31 
32 
33 
34 

35 
36 

39 
40 

41 
42 

43 
44 

t 

47 
48 

•57810 
•57833 

•57904 
•57928 

•81597 
•81580 

•81563 
•81546 
.81530 
•81513 

•59225 
.59248 
•59272 

•59295 
•59318 
59342 

.80576 
.80558 
.8054J 
.80524 
•80507 
•80489 

.60622 
.60645 
•60668 
•60691 
•60714 
•60738. 

•79530 
•79512 
•79494 
•79477 
•79459 
•79441 

.62001 
.62024 
.62046 
•62069 
•62092 
•62115 

•78460 
•78442 
.78424 
.78405 

•78387 
.78369 

.63361 

•63383 
.63406 
.63428 
•63451 
•63473 

.77366 

•77347 
.77329 
.77.310 
.77292 

•77273 

•57952 
•57976 
•57999 
•58023 

•58047 
•58070 

•81496 

81479 
.81462 
•  81445 
•81428 
.81412 

•593^5 
59389 
.59412 

•59436 

.80472 

•80455 
•80438 
.80420 
.80403 
•80386 

•60761 
•60784 
.60807 
.60830 
•60853 
•60876 

•79424 
.79406 
•79388 
•79371 
•79353 
•79335 
•79318 
.79300 
. 79282 
.79264 
.79247 
.79229 
.79211 

•79193 
.79176 

•79158 
.79140 
.79122 

•62138 
•62160 
.62183 
•62206 
.62229 
.62251 

•78351 
•78333 
•78315 
.78297 
.78279 
.78261 

•63496 
•63518 
•63540 
•63563 

•77255 
•77236 
.77218 

•77199 
.77181 
.77162 

.58141 
.58165 
.58189 
.58212 

•81395 
.81378 
.81361 

.81344 
.81327 
.81310 

•59506 
•59529 
•59552 
•59576 
•59599 
•59622 

•80368 
•80351 

.80264 
.80247 
•80230 
.80212 
•80195 
.80178 

•60899 
.60922 
•60945 
.60968 
.60991 
.61015 
•61038 
•61061 
.61084 
.61107 
.61130 
•61153 

.62274 
•62297 
•62320 
.62342 

^62388 

.78243 
.78225 
.78206 
.78188 
.78170 
.78152 

.63630 
•63653 

^63698 
.63720 
•63742 

•77144 
•77125 
.77107 
.77088 
.77070 
•77051 

29 

25 
24 

23 

22 
21 
20 
19 

.58236 
.58260 
•58283 
•58307 
•58330 
•58354 

.81293 
.81276 
.81259 
.81242 
•81225 
•81208 

•59646 
•59669 
•59693 
•59716 

•59739 
•59763 

.62411 
•62433 
•62456 
•62479 
•62502 
•62524 

•78134 
.78116 
.78098 

•78079 
.78061 

•78043 

•63765 
•63787 
.63810 
•63832 
•63854 
•63877 

•77033 
.77014 
.76996 
.76977 

•76959 
.76940 

•58378 
.58401 

•58425 
.58449 

.58472 
.58496 

•81191 
.81174 
.81157 
.81140 
.81123 
.81106 

•59786 
•59809 
59832 
•59856 
•59879 
•59902 

•80160 
•80143 
•80125 
•80108 
•80091 
■80073 

.61176 
.61199 
.61222 
•61245 
61268 
.61291 

•79105 
.79087 
.79069 
•79051 
•79033 
•79016 

•62547 
•62570 
•62592 
•62615 
•62638 
•62660 

.78025 
.78007 
•77988 
•77970 
•77952 
•77934 

•63899 
.63922 
.63944 
.63966 
.63989 
.64011 

.76921 
.76903 
.76884 
. 76866 
.76847 
•76828 

15 
14 
13 
12 

11 
10 

\ 

5 
4 
3 
2 
I 
0 

49 
50 

51 

52 
53 
54 

•58519 
•58637 

.81089 
.81072 
•81055 
.81038 
.81021 
.81004 

•59926 
•59949 
•59972 

•59995 
•60019 
.60042 

.80056 
.80038 
.80021 
•80003 
.79986 
•79968 

•79951 
•79934 
.79916 
•79899 
•79881 
•79864 

.61314 

.61383 

.61406 
.61429 
•61451 
.61474 
•61497 
•61520 

•61543 
.61566 

•78998 
•78980 
.78962 
.78944 
.78926 
.78908 

•62683 
.62706 
.62728 
•62751 
•62774 
•62796 

.77916 
•77897 
•77879 
.77861 

•77843 
.77824 

•64033 
•64056 
•64078 
.64100 
•64123 
•64145 

•76810 
•76791 
•76772 
•76754 
•76735 
.76717 

.76698 
.76679 
•76661 
•76642 
•76623 
.76604 

•58661 
.58684 
.58708 
58731 
•58755 
58779 

.80987 
.80970 

•80953 
•80936 
•80919 
•80902 

•60065 
•60089 
.60112 
•60135 
•60158 
•60182 

.78891 
•78873 
•78855 
•78837 
.78819 
.78801 

•62819 
.62842 
•62864 
•62887 
•62909 
•62932 

.77806 
.77788 
.77769 
•77751 

■nnz 
•77715 

•64167 
.64190 
.64212 
.64234 
.64256 
.64279 

N.  COS. 

N.  sine 

N.  COS.  ] 

V.  sine 

N.  COS. 

N^.  sine 

N.  COS.  ] 

V.  sme 

N^.  COS.  ] 

V.  sine 

54"    1 

53°    1 

52°    1 

51°    1 

50°    1 

80 


TABLE  III 


o 
I 

2 

3 

4 

1 

9 

lO 

II 

12 
33 

\l 

18 

40-    1 

41**    1 

42°    1 

«»     1 

44° 

60 

It 
% 

55 
54 

53 
52 
51 
50 

It 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

.64279 
.64301 

64323 
64346 
.64368 
.64390 
.64412 

.76604 
.76586 
.76567 
.76548 
•76530 
.76511 
.76492 

.65606 
.65628 
•65650 
.65672 

•65694 
.65716 

•65738 

•75471 
•75452 
•75433 
•75414 
•75395 
•75375 
•75356 

66913 
•66935 
.66956 
.66978 
.66999 
.67021 
•67043 
.67064 
.67086 
.67107 
.67129 
.67151 
.67172 

•74314 
•74295 
•74276 
•74256 
•74237 
•74217 
.74198 

.74178 
•74159 
•74139 
.74120 
.74100 
.74080 

.68200 
.68221 
.68242 
.68264 
.68285 
.68306 
•68327 

•73135 
.73116 
.73096 
■73076 
•73056 
•73036 
.73016 

.69466 

•69487 
.69508 

•69529 
•69549 
.69570 
.69591 

•71934 
.71914 
.71894 
■71873 
•71853 
•71833 
.71813 

•64435 
•64457 
•64479 
.64501 
.64524 
.64546 

•76473 
■76455 
.76436 
.76417 
.76398 
.76380 

•65759 

•65803 
.65825 
.65847 
.65869 

■7S337 
•75318 

•75299 
.75280 

•75261 

•75241 

•68349 

.68370 

•68391 

.68412 

•68434 

•68455 

.68476" 

•68497 

.68518 

•68539 
.68561 
.68582 

.72996 
.72976 

•72957 
.72937 
.72917 
.72897 

.69612 
.69633 
.69654 

•69675 
.69696 
.69717 

.71792 
.71772 
•71752 
.71732 
.71711 
.71691 

.64568 
.64590 
.64612 

■64635 
.64657 
.64679 

•76361 
•76342 
•76323 
.76304 
.76286 
.76267 

.65891 
•65913 

•65956 
.65978 
.66000 

.75222 
•75203 
•75184 
•75165 
•75146 
.75126 

.67194 
.67215 

•67237 
.67258 
.67280 
.67301 

.74061 
.74041 
.74022 
.74002 
•73983 
•73963 

.72877 
.72857 

•72837 
.72817 

•72797 

■72777 

•69737 
•69758 
.69779 
.69800 
.69821 
.69842 

.71671 
.71650 
.71630 
.71610 
•71590 
•71569 

% 

45 
44 
43 
42 

19 
20 
21 
22 

23 

24 

11 
11 

29 
3c 

31 
32 

33 

34 
35 
36 

11 

39 
40 

41 
42 

43 
44 

:i 

47 
48 

49 
50 
51 
52 
53 
54 

55 
56 

11 

.64701 
.64723 
.64746 
.64768 
.64790 
.64812 

.76248 
.76229 
.76210 
.76192 
.76173 
•76154 

•76135 
.76116 
.76097 
.76078 

•76059 
.76041 

.76022 
.76003 
•75984 
•75965 
•75946 
•75927 
.75908 
.75889 
•75870 
.75851 
•75832 
•75813 

.66022 
.66044 
.66066 
.66088 
.66109 
.66131 

.75069 
•75050 
•75030 
.75011 

•67323 
67344 
•67366 

.67409 
•67430 

•73944 
•73924 
•73904 
■73885 
•73865 
.73846 

.68603 
.68624 
.68645 
.68666 
.68688 
.68709 

■72757 
■72737 
•72717 
.72697 
.72677 
.72657 

.69862 
•69883 
.69904 

.69946 
.69966 

•71549 
•71529 
.71508 
.71488 
.71468 
•71447 

41 
40 

35 
34 
33 
32 
31 
30 

29 
28 

% 

25 
24 

23 
22 
21 

20 
19 

\l 

15 
14 
13 
12 

II 
10 

i 

7 

6 

.64834 
64856 
64878 
.64901 
64923 
64945 
64967 
.64989 
.65011 

•65033 
■65055 
■65077 
.65100 
.65122 
.65144 
.65166 
.65188 
.65210 

66175 
.66197 
.66218 
.66240 
.66262 

•74992 
•74973 
•74953 
•74934 

•74896 

.67452 
■67473 

■67538 
•67559 

.73826 
•73806 
•73787 
•73767 
•73747 
•73728 

•68730 
.68751 
.68772 

68793 
.68814 

■68835, 

.72637 
.72617 
•72597 
•72577 
•72557 
•72537 

•69987 
.70008 
.70029 
.70049 
.70070 
.70091 

.71427 
.71407 
.71386 
.71366 
■71345 
■71325 

.66284 
.66306 
•66327 
•66349 

•66393 
.66414 
.66436 
.66458 
.66480 
.66501 
•66523 

.74876 
•74857 

■.Ifsfs 

.67580 
.67602 
.67623 

.67688 

.73708 
•73688 
73669 
•73649 
.73629 
.73610 

.68857 
.68878 
.68899 
.68920 
.68941 
.68962 

•72517 
.72497 

•72477 
•72457 
•72437 
.72417 

.70112 
.70132 

•70153 
.70174 

•70195 
•70215 

•71305 
.71284 
.71264 

•71243 
.71223 
.71203 

.74760 

•74741 
.74722 

•74703 
■74683 
.74664 

.67709 
.67730 
.67752 
•67773 
•67795 
.67816 

•73590 
•73570 
•73551 
•73531 
•735" 
•73491 

.68983 
.69004 
.69025 
.69046 
.69067 
.69088 

•72397 
■72377 
■72357 
■72337 
.72317 
.72297 

.70236 
•70257 
.70277 
.70298 
.70319 
•70339 

.71182 
.71162 
.71141 
.71121 

.71100 
.71080 

.65232 

•65254 
•65276 
.65298 
•65320 
•65342 

•75794 
•75775 
•75756 
•75738 
•75719 
■75700 

"66566 
66588 
.66610 
.66632 
•66653 

.74644 

.74606 
•74586 
•74567 
•74548 

•67837 

.67901 

•67923 
.67944 

•73472 
•73452 
•73432 
•73413 
•73393 
■73373 

.69109 
.69130 
.69151 
.69172 
.69193 
.69214 

.72277 

•72257 
.72236 
.72216 
.72196 
.72176 

.70360 
.70381 
.70401 
.70422 
•70443 
•70463 

•71059 
.71039 
.71019 
.70998 
•70978 
•70957 

•65364 
.65386 
.65408 
•65430 
•65452 
•65474 

.75680 
.75661 
•75642 
•75623 
.75604 

•75585 

.66675 
.66697 
.66718 
.66740 
.66762 
•66783 

.74528 
•74509 
•74489 
•74470 
•74451 
•74431 

.67965 

.68029 
.68051 
.68072 

•73353 
•73333 
•73314 
•73294 
•73274 
•73254 

•69235 
.69256 
.69277 
.69298 
.69319 
.69340 

.72156 
•72136 
.72116 
.72095 
•72075 
•72055 

.70484 
•70505 
•70525 
.70546 

•70567 
•70587 

.70937 
.70916 
.70896 
.70875 
•70855 
.70834 

•65496 
.65518 
•65540 
65562 
.65584 
.65606 

•75566 
•75547 
•75528 
•75509 
•75490 
•75471 

.66805 
.66827 
.66848 
.66870 
.66891 
•66913 

.74412 
•74392 

•74373 
•74353 
•74334 
•74314 

.68093 
.68115 
.68136 
.68157 
.68179 
.68200 

•73234 
•73215 
•73195 
•73175 
•73155 
•73135 

•69361 
.69382 
.69403 
.69424 

•72035 
•72015 
•71995 
•71974 
•71954 
71934 

.70608 
.70628 
.70649 
.70670 
.70690 
.70711 

.70813 

•70793 
.70772 

•70752 
•70731 
.70711 

5 
4 
3 

2 
I 
0 

t 

N.  COS.  ] 

NT.  sine 

f^.  COS.  ] 

N".  sine 

NT.  COS.  ] 

N".  sine 

^J.  CO.S.  ^ 

ST.  sine 

N'.  cos.  ] 

V.  sine 

49°    1 

4§*'    1 

47°    1 

46°    1 

45-    1 

NATURAL  TANGENTS  AND  COTANGENTS 


81 


/ 

0^^ 

1° 

2^" 

3° 

40 

/ 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

0 

I 

2 

3 
4 

0000  Infinite 
0003  3437-75 
0006  1718.87 
0009  1145.92 
0012  859.436 

0175  57.2900 
0177  56.3506 
0180  55.4415 
0183  54.5613 
0186  53.7086 

0349  28.6363 
0352  28.3994 
0355  28.1664 
0358  27.9372 
0361  27.7117 

0524  19.0811 
0527  18.9755 
0530  18.8711 
0533  18.7678 
0536  18.6656 

0699  14.3007 
0702  14.2411 
070^  14.1821 
0708  14.1235 
0711  14.0655 

60 

5 
6 

7 
8 

9 

0015  687.549 
0017  572957 
0020  491.106 
0023  429.718 
0026  381.971 

0189  52.8821 
0192  52.0807 
0195  51.3032 
0198  50.5485 
0201  49.8157 

0364  27.4899 
0367  27.2715 
0370  27.0566 
0373  26.8450 
0375  26.6367 

0539  18.5645 
0542  18.4645 
0544  18.3655 
0547  18.2677 
0550  18.1708 

0714  14.0079 
0717  13-9507 
0720  13.8940 

0723  138378 
0726  13.7821 

55 
54 
53 
52 
51 

10 

II 

12 

13 
14 

0029  343-774 
0032  312.521 
0035  286.478 
0038  264.441 
0041  245.552 

0204  49.1039 
0207  48.4121 
0209  47.7395 
0212  47.0853 
0215  46.4489 

0378  26.4316 
0381  26.2296 
0384  26.0307 
0387  25.8348 
0390  25.6418 

0553  18.0750 
0556  17.9802 
OS59  17.8863 
0562  17.7934 
^565  17-7015 

0729  13.7267 
0731  13.6719 
0734  13-6174 
0737  13-5634 
0740  13.5098 

50 

49 
48 
47 
46 

17 
i8 

19 

0044  229.152 
0047  214.858 
0049  202,219 
0052  190.984 
0055  180.932 

0218  45.8294 
0221  45.2261 
0224  44.6386 
0227  44.0661 
0230  43.5081 

0393  25.4517 
0396  25.2644 
0399  25.0798 
0402  24.8978 
0405  24.7185 

0568  17.6106 
0571  17.5205 
0574  1 7.43  H 
0577  17.3432 
0580  17.2558 

0743  13.4566 
0746  13.4039 
0749  13.3515 
0752  13.2996 
0755  13.2480 

45 
44 
43 
42 
41 

20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

0058  171.885 
0061  163.700 
0064  156.259 
0067  149.465 
0070  143.237 

0233  42.9641 
0236  42.4335 
0239  41-9158 
0241  41.4106 
0244  40.9174 

0407  24.5418 
0410  23.3675 
0413  24.1957 
0416  24.0263 
0419  23.8593 

0582  17.1693 
0585  17.0837 
0588  16.9990 
0591  16.9150 
0594  16.8319 

0758  13.1969 
0761  13.1461 
0764  13.0958 
0767  13.0458 
0769  12.9962 

40 

39 
38 
31 
36 

0073  137-507 
0076  132.219 
0079  127.321 
0081  122.774 
0084  118.540 

0247  40.4358 
0250  39.9655 
0253  39.5059 
0256  39.0568 
0259  38.6177 

0422  23.6945 
0425  23.5321 
0428  23.3718 
0431  23.2137 
0434  23.0577 

0597  16.7496 
0600  16.6681 
0603  16.5874 
0606  16.5075 
0609  16.4283 

0772  12.9469 
0775  12.8981 
0778  12.8496 
0781  12.8014 
0784  12.7536 

35 
34 
32, 
32 
31 

30 

31 

32 

34 

0087  114-589 
0090  110.892 
0093  107.426 
0096  104.171 
0099  101.107 

0262  38.1885 
0265  37.7686 
0268  37-3579 
0271  36.9560 
0274  36.5627 

0437  22.9038 
0440  22.7519 
0442  22.6020 
0445  22.4541 
0448  22.3081 

0612  16.3499 
0615  16.2722 
0617  16.1952 
0620  16.1190 
0623  16.0435 

0787  12.7062 
0790  12.6591 
0793  12.6124 
0796  12.5660 
0799  12.5199 

30 

29 
28 

27 
26 

35 
36 

39 

0102  98.2179 
0105  95.4895 
0108  92.9085 
oiii  904633 
01 13  88.1436 

0276  36.1776 
0279  35.8006 
0282  35-4313 
0285  35-0695 
0288  34.7151 

0451  22.1640 
0454  22.0217 
0457  21.8813 
0460  21.7426 
0463  21.6056 

0626  15.9687 
0629  15.8945 
0632  15.8211 

0635  15.7483 
0638  15.6762 

0802  12.4742 
0805  12.4288 
0808  12.3838 
0810  12.3390 
0813  12.2946 

25 
24 

23 
22 
21 

40 

41 

42 

43 
44 

01 16  85.9398 
01 19  83.8435 
0122  81.8470 
0125  79.9434 
0128  78.1263 

0291  34.3678 
0294  34.0273 

0297  336935 
0300  33.3662 

0303  33-0452 

0466  21.4704 
0469  21.3369 
0472  21.2049 
0475  21.0747 
0477  20.9460 

0641  15.6048 
0644  15-5340 
0647  15.4638 
0650  15.3943 
0653  15.3254 

0816  12.2505 
0819  12.2067 
0822  12.1632 
0825  12.1201 
0828  12.0772 

20 

19 

18 

17 
i6 

45 
46 

47 
48 

49 

0131  76.3900 
0134  74.7292 
0137  73.1390 
0140  71.6151 
0143  70.1533 

0306  32.7303 
0308  32.4213 
0311  32.1181 
0314  31.8205 
0317  31.5284 

0480  20.8188 
0483  20.6932 
0486  20.5691 
0489  20.4465 
0492  20.3253 

0655  15.2571 
0658  15.1893 
0661  15.1222 
0664  15.0557 
0667  14.9898 

0831  12.0346 
0834  11.9923 
0837  11.9504 
0840  11.9087 
0843  11.8673 

15 
14 
13 
12 
11 

50 

51 

52 
53 
54 

59 

0146  68.7501 
0148  67.4019 
0151  66.1055 
0154  64.8580 
0157  63.6567 

0320  31.2416 

0323  30.9599 
0326  30.6833 
0329  30.4116 
0332  30.1446 

0495  20.2056 
0498  20.0872 
0501  19.9702 
0504  19.8546 
0507  19.7403 

0670  14.9244 
0673  14-8596 
0676  14.7954 
0679  14-7317 
0682  14.6685 

0846  11.8262 
0849  11.7853 
0851  11.7448 
0854  11.7045 
0857  11.6645 

10 

I 

7 
6 

5 
4 
3 
2 

I 

0160  62.4992 
0163  61,3829 
0166  60.3058 
0169  59.2659 
0172  58.2612 

0335  29.8823 
0338  29.6245 
0340  29.3711 
0343  29.1220 
0346  28.8771 

0509  19.6273 

OCil2  19.5156 
0515  19.4051 
0518  19.2959 
0521   19.1879 

0685  14.6059 
0688  14.5438 
0690  14.4823 
0693  14.4212 
0696  14.3607 

0860  11.6248 
0863  11.5853 
0866  11.5461 
0869  11.5072 
0872  11.4685 

60 

0175  57.2900 

0349  28.6363 

0524  19.0811 

0699  14.3007 

0875  11.4301 

0 

(.otg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

/ 

8^0 

8H0 

H7^    i    H(i^ 

85^ 

/ 

82 


TABLE   III 


/ 

50 

0° 

7" 

8'-^ 

90 

/ 

Tang  Cotg 

Tang 

totg 

Tang  Cotg 

Tang  Cotg 

Tang 

Cotg 

60 

0 

0875  1 1. 4301 

1051 

9.5144 

1228  8.1443 

1405  7.1154 

1584 

6.3138 

I 

0878  11.3919 

1054 

9-4878 

1231  8.1248 

1408  7.1004 

1587 

6.3019 

59 

2 

0881  11.3540 

1057 

9.4614 

1234  8.1054 

141 1  7.0855 

1590 

6.2901 

58 

^ 

0884  1 1. 3 1 63 

1060 

94352 

1237  8.0860 

1414  7.0706 

1593 

6.2783 

57 

4 

0887  11.2789 

1063 

9.4090 

1240  8.0667 

1417  7-0558 

1596 

6.2666 

56 

s 

0890  1 1.24 1 7 

1066 

9-3831 

1243  8.0476 

1420  7.0410 

1599 

6.2549 

55 

6 

0892  11.2048 

1069 

9-3572 

1246  8.0285 

1423  7.0264 

1602 

6.2432 

54 

7 

0895  1 1. 1 681 

1072 

9-3315 

1249  8.0095 

1426  7.0117 

1605 

6.2316 

53 

8 

0898  11.1316 

1075 

9.3060 

1 25 1  7.9906 

1429  6.9972 

1608 

6.2200 

52 

9 

0901  11.0954 

1078 

9.2806 

1254  7.9718 

1432  6.9827 

1611 

6.2085 

51 

10 

0904  11.0594 

1080 

9-2553 

1257  7.9530 

1435  6.9682 

1614 

6.1970 

50 

II 

0907  11.0237 

1083 

9.2302 

1260  7.9344 

1438  6.9538 

1617 

6.1856 

49 

12 

9910  10.9882 

1086 

9.2052 

1263  7.9158 

1441  6.9395 

1620 

6.1742 

48 

1,3 

0913  10.9529 

1089 

9.1803 

1266  7.8973 

1444  6.9252 

1623 

6.1628 

47 

14 

0916  10.9178 

1092 

9-1555 

1269  7.8789 

1447  6.91 10 

1626 

6.1515 

46 

IS 

0919  10.8829 

1095 

9.1309 

1272  7.8606 

1450  6.8969 

1629 

6.1402 

45 

i6 

0922  10.8483 

1098 

9.1065 

1275  7.8424 

1453  6.8828 

1632 

6.1290 

44 

17 

0925  10.8139 

IIOI 

9.0821 

1278  7.8243 

1456  6.8687 

1635 

6.1178 

43 

18 

0928  10.7797 

1104 

9-0579 

1281  7.8062 

1459  6.8548 

1638 

6.1066 

42 

19 

0931  10.7457 

1 107 

9-0338 

1284  7.7883 

1462  6.8408 

1641 

6-0955 

41 

20 

0934  10.71 19 

mo 

9.0098 

1287  7.7704 

1465  6.8269 

1644 

6.0844 

40 

21 

0936  10.6783 

1113 

8.9860 

1290  7.7525 

1468  6.8131 

1647 

6.0734 

39 

22 

0939  10.6450 

1116 

8.9623 

1293  7.7348 

1471  6.7994 

1650 

6.0624 

38 

23 

0942  10.61 18 

1119 

8.9387 

1296  7.7171 

1474  6.7856 

1653 

6.0514 

37 

24 

0945  10.5789 

1122 

8.9152 

1299  7.6996 

1477  6.7720 

1655 

6.0405 

36 

2S 

0948  10.5462 

1125 

8.8919 

1302  7.6821 

1480  6.7584 

1658 

6.0296 

35 

26 

0951  10.5136 

1128 

8.8686 

1305  7.6647 

1483  6.7448 

1661 

6.0188 

34 

27 

0954  10.4813 

1131 

8.8455 

1308  7.6473 

i486  6.7313 

1664 

6.0080 

33 

28 

0957  10.4491 

1 134 

8.8225 

1 31 1  7.6301 

1489  6.7179 

1667 

5-9972 

32 

29 

0960  10.4172 

1 136 

8.7996 

1314  7.6129 

1492  6.7045 

1670 

5.9865 

31 

30 

0963  10.3854 

1 139 

8.7769 

1317  7-5958 

1495  6.0912 

Tell 

5-9758 

30 

31 

0966  10.3538 

1 142 

8.7542 

1319  7.5787 

1497  6.6779 

5-9651 

29 

32 

0969  10.3224 

1 145 

8.7317 

1322  7.5618 

1500  6.6646 

1 679 

5.9545 

28 

33 

0972  10,2913 

1148 

8.7093 

1325  7.5449 

1503  6.6514 

1682 

5.9439 

27 

34 

0975  10.2602 

1151 

8.6870 

1328  7.5281 

1506  6.6383 

1685 
1688 

5-9333 
5.9228 

26 
25 

3S 

0978  10.2294 

1154 

8.6648 

1331  7.5113 

1509  6.6252 

36 

0981  10.1988 

1157 

8.6427 

1334  7.4947 

1512  6.6122 

1691 

5.9124 

24 

37 

0983  10.1683 

1 160 

8.6208 

1337  7.4781 

1515  6.5992 

1694 

5-9019 

23 

38 

0986  10.1381 

1163 

8.5989 

1340  7.4615 

15 18  6.5863 

1697 

5.8915 

22 

39 

0989  10.1080 

1166 

8-5772 

1343  7.4451 

1521  6.5734 

1700 

5.8811 

21 

40 

0992  10.0780 

1 169 

8.5555 

1346  7.4287 

1524  6.5606 

1703 

5.8708 

20 

41 

0995  10.0483 

1172 

8.5340 

1349  7.4124 

1527  6.5478 

1706 

5.8605 

19 

42 

0998  10.0187 

1 175 

8.5126 

1352  7.3962 

1530  6.5350 

1709 

5.8502 

18 

43 

looi  9.9893 

1178 

8.4913 

1355  7.3800 

1533  6.5223 

1712 

5.8400 

17 

44 

1004  9.9601 

8.4701 

1358  7.3639 

1536  6.5097 

1715 

6.8298 

16 
15 

45 

1007  9.9310 

1 184 

8.4490 

1361  7.3479 

1539  6.4971 

1718 

58197 

46 

loio  9.9021 

1187 

8.4280 

1364  7.3319 

1542  6.4846 

1721 

5-8095 

14 

47 

IOI3  9-8734 

1 189 

8.4071 

1367  7.3160 

1545  6.4721 

1724 

5-7994 

13 

48 

1016  9.8448 

1192 

8.3863 

1370  7.3002 

1548  6.4596 

1727 

5-7894 

12 

49 

IOI9  9.8164 

"95 

8.3656 

1373  7.2844 

1551  6.4472 

1730 

5-7794 

II 

50 

1022  9.7882 

1 198 

8.3450 

1376  7.2687 

1554  6.4348 

1733 

5-7694 

10 

SI 

1025  9.7601 

1 201 

8.3245 

1379  7.2531 

1557  6.4225 

1736 

5-7594 

9 

S2 

1028  9-7322 

1204 

8.3041 

1382  7.2375 

1560  6.4103 

1739 

5-7495 

8 

S3 

1030  9.7044 

1207 

8.2838 

1385  7.2220 

1563  6.3980 

1742 

5.7396 

7 

54 

1033  9-6768 

1210 

8.2636 

1388  7.2066 

1566  6.3859 

1745 

5.7297 

6 

5S 

1036  9-6499 

1213 

8.2434 

1391  7.1912 

1569  6.3737 

1748 

5.7199 

5 

5^ 

1039  9.6220 

1216 

8.2234 

1394  7.1759 

1572  6.3617 

1751 

5.7101 

4 

S7 

1042  9-5949 

1219 

8.2035 

1397  7.1607 

1575  6.3496 

1754 

5.7004 

3 

5« 

1045  9.5679 

1222 

8.1837 

1399  7.1455 

1578  6.3376 

1757 

5.6906 

2 

59 

1048  9-541 1 

1225 

8.1640 

1402  7.1304 

1581  6.3257 

1760 

5.6809 

I 

00 

1051  9-5144 

1228 

8.1443 

1405  7.1 154 

1584  6.3138 
Cotg  Tang 

1763 
Cotg 

5-6713 
Tang 

0 

Cotg  Tang 

Cotg 

Tang 

Cotg  Tang 

L. 

84° 

83° 

82^ 

81" 

80" 

/ 

NATURAL  TANGENTS  AND  COTANGENTS 

83 

/ 

10^    1 

11^ 

12° 

13° 

14° 

/ 

Taug 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

60 

0 

^I^Z 

5-6713 

1944 

5.1446 

2126 

4.7046 

2309 

4-3315 

2493 

4.0108 

I 

1766 

5.6617 

1947 

5.1366 

2129 

4-6979 

2312 

4-3257 

2496 

4.0058 

5? 

2 

1769 

5-6521 

1950 

5.1286 

2132 

4.6912 

2315 

4.3200 

2499 

4.0009 

58 

^ 

1772 

5-6425 

1953 

5.1207 

2135 

4.6845 

2318 

4-3143 

2503 

3-9959 

57 

4 

1775 

5-6330 

1956 

5.1128 

2138 

4.6779 

2321 

4.3086 

2506 

3.9910 

5^ 

S 

1778 

5-6234 

1959 

5-I049 

2141 

4.6712 

2324 

4-3029 

2509 

3-9861 

55 

6 

I78I 

5.6140 

1962 

5-0970 

2144 

4.6646 

2327 

4-2972 

2512 

3-9812 

54 

7 

1784 

5.6045 

1965 

5.0892 

2147 

4.6580 

2330 

4.2916 

2515 

3-9763 

53 

8 

1787 

5-5951 

1968 

5.0814 

2150 

4.6514 

233.3 

4-2859 

2518 

3-9714 

52 

9 
10 

1790 

5-5857 

1971 

5-0736 

2153 

4.6448 

2336 

4.2803 

2521 

3.9665 

51 

1793 

5-5764 

1974 

5.0658 

2156 

4.6382 

2339 

4.2747 

2524 

3-9617 

50 

II 

1796 

5-5671 

1977 

5-0581 

2159 

4-6317 

2342 

4.2691 

2527 

3-9568 

49 

12 

1799 

5-5578 

1980 

5-0504 

2162 

4.6252 

2345 

4-2635 

2530 

3.9520 

48 

i,S 

1802 

5-5485 

1983 

5.0427 

2165 

4.6187 

2349 

4.2580 

2533 

3-9471 

47 

14 

1805 

5-5393 

1986 

5-0350 

2168 

4.6122 

2352 

4.2524 

2537 

3-9423 

46 

IS 

1808 

5-5301 

1989 

5-0273 

2171 

4-6057 

2355 

4.2468 

2540 

3-9375 

45 

i6 

I8II 

5-5209 

1992 

50197 

2174 

4.5993 

2358 

4.2413 

2543 

3-9327 

44 

17 

I8I4 

5.5118 

1995 

5.0121 

2177 

4-5928 

2361 

4.2358 

2546 

3-9279 

43 

18 

I8I7 

5.5026 

1998 

5-0045 

2180 

4-5864 

2364 

4.2303 

2549 

3-9232 

42 

19 

1820 

5-4936 

2001 

4.9969 

2183 

4.5800 

2367 

4.2248 

2552 

3.9184 

41 

20 

1823 

5-4845 

2004 

4.9894 

2186 

4-5736 

2370 

4.2193 

2555 

3-9136 

40 

21 

1826 

5-4755 

2007 

4.9819 

2189 

4.5673 

2373 

4.2139 

2558 

3.9089 

39 

22 

1829 

5-4665 

2010 

4.9744 

2193 

4.5609 

2376 

4.2084 

2561 

3.9042 

38 

23 

1832 

5-4575 

2013 

4.9669 

2196 

4-5546 

2379 

4.2030 

2564 

3-8995 

31 

24 

i«35 

5-4486 

2016 

4-9594 

2199 

4.5483 

2382 

4.1976 

2568 

3-8947 

36 

2S 

1838 

5-4397 

2019 

4.9520 

2202 

4.5420 

2385 

4.1922 

2571 

3.8900 

35 

26 

1841 

5-4308 

2022 

4-9446 

2205 

4.5357 

2388 

4.1868 

2574 

3-8854 

34 

27 

1844 

5-4219 

2025 

4-9372 

2208 

4.5294 

2392 

4.1814 

2577 

3-8807 

33 

28 

1847 

5-4131 

2028 

4.9298 

221 1 

4.5232 

2395 

4.1760 

2580 

3-8760 

32 

29 

1850 

5-4043 

2031 

4.9225 

2214 

4.5169 

2398 

4.1706 

2583 

3-8714 

31 

30 

i8S3 

5-3955, 

2035 

4.9152 

2217 

4-5107 

2401 

4.1653 

2586 

38667 

30 

31 

1856 

5.3868 

2038 

4.9078 

2220 

4.5045 

2404 

4.1600 

2589 

3.8621 

29 

32 

i«59 

5-3781 

2041 

4.9006 

2223 

4.4983 

2407 

4-1547 

2592 

3-8575 

28 

33 

1862 

5-3694 

2044 

4-8933 

2226 

4.4922 

2410 

4.1493 

2.595 

3-8528 

27 

34 

1865 

5-3607 

2047 

4.8860 

2229 

4.4860 

2413 

4.1441 

2599 

3.8482 

26 

35 

1868 

5-3521 

2050 

4.8788 

2232 

4.4799 

2416 

4.1388 

2602 

3-8436 

25 

3^ 

1871 

5-3435 

2053 

4.8716 

2235 

4.4737 

2419 

4.1335 

2605 

3-8391 

24 

37 

1874 

5-3349 

2056 

4.8644 

2238 

4.4676 

2422 

4.1282 

2608 

3-8345 

23 

38 

1877 

5-3263 

2059 

4.8573 

2241 

4-4615 

2425 

4.1230 

261 1 

3-8299 

22 

39 

1880 

5-3178 

2062 

4-8501 

2244 

4-4555 

2428 

4.1178 

2614 

3-8254 

21 

40 

1883 

5-3093 

2065 

4-8430 

2247 

4.4494 

2432 

4.1126 

2617 

3.8208 

20 

41 

1887 

5-3008 

2068 

4-8359 

2251 

4.4434 

2435 

4.1074 

2620 

3-8163 

19 

42 

1890 

5-2924 

2071 

4.8288 

2254 

4-4374 

2438 

4.1022 

2623 

3.81 18 

18 

43 

1893 

5-2839 

2074 

4.8218 

2257 

4.4313 

2441 

4.0970 

2627 

3-8073 

17 

44 
45 

1896 
1899 

5-2755 

2077 

4.8147 

2260 

4-4253 

2444 

4.0918 

2630 

3.8028 

16 

5.2672 

2080 

4-8077 

2263 

4.4194 

2447 

4.0867 

2633 

3-7983 

15 

4b 

1902 

5.2588 

2083 

4.8007 

2266 

4-4134 

2450 

4-0815 

2636 

3-7938 

14 

47 

1905 

5-2505 

2086 

4-7937 

2269 

4.4075 

2453 

4.0764 

2639 

3-7893 

13 

48 

1908 

5-2422 

2089 

4.7867 

2272 

4.4015 

2456 

4.0713 

2642 

3-7848 

12 

49 

1911 

5-2339 

2092 

4.7798 

2275 

4.3956 

2459 

4.0662 

2645 

3.7804 

II 

50 

1914 

5-2257 

2095 

4-7729 

2278 

4.3897 

2462 

4.061 1 

2648 

3.7760 

10 

51 

1917 

5-2174 

2098 

4-7659 

2281 

4.3838 

2465 

4.0560 

2651 

3-7715 

9 

52 

1920 

5.2092 

2101 

4-7591 

2284 

4.3779 

2469 

4.0509 

2655 

3-7671 

8 

53 

1923 

5.2C1 1 

2104 

4.7522 

2287 

4-3721 

2472 

4.0459 

2658 

3.7627 

7 

54 
55 

1926 

5.1929 

2107 

4-7453 

2290 

4.3662 

2475 

4.0408 

2661 

3-7583 

b 

1929 

5.1848 

2110 

4.7385 

2293 

4.3604 

2478 

4.0358 

2664 

3-7539 

'5 

5<^ 

1932 

^•^lll 

2113 

4-7317 

2296 

4-3546 

2481 

4.0308 

2667 

3-7495 

4 

57 

1935 

5.1686 

2116 

4.7249 

2299 

4-3488 

2484 

4.0257 

2670 

3-7451 

3 

5« 

i93« 

5.1606 

2119 

4.7181 

2303 

4.3430 

2487 

4.0207 

2673 

3.7408 

2 

59 

1 941 

5-1526 

2123 

4.7114 

2306 

4.3372 

2490 

4.0158 

2676 

3.7364 

I 

00 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

2679 

3-7321 

0 

Cotg 

Tan^ 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

/ 

7 

0° 

78° 

77  ' 

7 

6^ 

7 

5° 

f 

84 


TABLE  III 


/ 

15° 

1(>° 

17° 

18^ 

193 

/ 

ian^ 

Cot^ 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

0 

2679 

3.7321 

2867 

3-4874 

3057 

3-2709 

3249 

3.0777 

3443 

2.9042 

60 

I 

2683 

3-7277 

2871 

34836 

3060 

3-2675 

3252 

3.0746 

3447 

2.9015 

S9 

2 

2686 

3-7234 

2874 

3-4798 

3064 

3.2641 

3256 

3.0716 

3450 

2.8987 

S8 

3 

2689 

3-7191 

2877 

3.4760 

3067 

3.2607 

3259 

3.0686 

3453 

2.8960 

S7 

4 

2692 

3-7148 

2880 

3.4722 

3070 

3.2573 

3262 

3.0655 

3456 

2-8933 

'^6 

5 

2695 

3-7105 

2883 

3-4684 

.3073 

32539 

3265 

3.0625 

3460 

2.8905 

SS 

6 

2698 

3.7062 

2886 

3-4646 

3076 

3.2506 

3269 

3.0595 

3463 

2.8878 

S4 

7 

2701 

3-7019 

2890 

3.4608 

3080 

3-2472 

3272 

3.0565 

3466 

2.8851 

S3 

8 

2704 

3.6976 

2893 

34570 

.3083 

3.2438 

3275 

3.0535 

3469 

2.8824 

S2 

9 

2708 

3-6933 

2896 

3-4533 

3086 

3-2405 

3278 

3-0505 

3473 

2.8797 

51 

50 

10 

2711 

3.6891 

2899 

3-4495 

3089 

3-2371 

3281 

3.0475 

.3476 

2.8770 

II 

2714 

3.6848 

2902 

3-4458 

3092 

3-2338 

3285 

3.0445 

3479 

2.8743 

49 

12 

2717 

3.6806 

2905 

3.4420 

3096 

3.2305 

3288 

3.0415 

.3482 

2.8716 

48 

1.3 

2720 

3.6764 

2908 

3-4383 

3099 

3.2272 

3291 

30385 

.3486 

2.8689 

47 

14 

2723 

3.6722 

2912 

3-4346 

3102 

3-2238 

3294 

3-0356 

3489 

2.8662 

46 

15 

2726 

3.6680 

2915 

3-4308 

3105 

3.2205 

3298 

3.0326 

3492 

2.8636 

4S 

lb 

2729 

3.6638 

2918 

3-4271 

3108 

3.2172 

3301 

3.0296 

3495 

2.8609 

44 

17 

2733 

3.6596 

2921 

3-4234 

3111 

3.2139 

3304 

3.0267 

3499 

2.8582 

43 

18 

2736 

3-6554 

2924 

3-4197 

3"5 

3.2106 

3307 

3.0237 

3502 

2.8556 

42 

19 

2739 

3.6512 

2927 

3.4160 

3118 

3.2073 

3310 

3.0208 

3505 

2.8529 

41 

20 

2742 

3.6470 

2931 

3.4124 

3121 

3.2041 

3314 

3.0178 

3So8 

2.8502 

40 

21 

2745 

3.6429 

2934 

3.4087 

3124 

3.2008 

3317 

3.0149 

3512 

2.8476 

39 

22 

2748 

3.6387 

2937 

3.4050 

3127 

3.1975 

3320 

3.0120 

3515 

2.8449 

38 

23 

2751 

3-6346 

2940 

3-4014 

3131 

3.1943 

3323 

3.0090 

3518 

2.8423 

37 

24 

2754 

3-6305 

2943 

3-3977 

3134 

3.1910 

3327 

3.0061 

3522 

2.8397 

36 

25 

2758 

3.6264 

2946 

3-3941 

3137 

3.1878 

3330 

3.0032 

.3525 

2.8370 

3S 

26 

2761 

3.6222 

2949 

3-3904 

3140 

3.1845 

3333 

3.0003 

3528 

2.8344 

34 

27 

2764 

3.6181 

29S3 

3-3868 

3143 

3-1813 

3336 

2.9974 

3S3I 

2.8318 

33 

28 

2767 

3.6140 

2956 

3.3832 

3147 

3.1780 

3339 

2-9945 

3535 

2.8291 

32 

29 

2770 

3.6100 

2959 

3-3796 

3150 

3-1748 

3343 

2.9916 

3538 

2.8265 

31 

30 

2773 

3-6059 

2962 

3.3759 

3153 

3.1716 

3346 

2.9887 

3541 

2.8239 

30 

31 

2776 

3.6018 

2965 

3.3723 

3156 

3.1684 

3349 

2.9858 

3S44 

2.8213 

29 

32 

2780 

3-5978 

2968 

3.3687 

3I.S9 

3-1652 

3352 

2.9829 

3548 

2.8187 

28 

^^ 

2783 

3-5937 

2972 

3.3652 

3163 

3.1620 

3.356 

2.9800 

3551 

2.8161 

27 

34 

2786 

3.5897 

2975 

3-3616 

3166 

3-1588 

3359 

2.9772 

3554 

2.8135 

26 

35 

2789 

3-5856 

2978 

3-3580 

3169 

3-1556 

3362 

2.9743 

3558 

2.8109 

25 

3b 

2792 

3.5816 

2981 

3.3544 

3172 

3.1524 

3365 

2.9714 

3561 

2.8083 

24 

37 

2795 

3-5776 

2984 

3.3509 

3175 

3.1492 

3369 

2.9686 

3564 

2.8057 

23 

38 

2798 

3-5736 

2987 

3-3473 

3179 

3.1460 

.3372 

2.9657 

3567 

2.8032 

22 

39 

40 

2801 

3.5696 

2991 

3.3438 

3182 

3.1429 

3375 

2.9629 

3571 

2.8006 

21 

2805 

3.5656 

2994 

3.3402 

318s 

3.1397 

3378 

2.9600 

3574 

2.7980 

20 

41 

2808 

3-5616 

2997 

3.3367 

3188 

3.1366 

3382 

2.9572 

3577 

2.7955 

19 

42 

2811 

3-5576 

3000 

3-3332 

3191 

3.1334 

3385 

2.9544 

3581 

2.7929 

18 

43 

2814 

3-5536 

3003 

3.3297 

3195 

3.1303 

3388 

2.9515 

3584 

2.7903 

17 

44 

2817 

3-5497 

3006 

3.3261 

3198 

3.1271 

3391 

2.9487 

3587 

2.7878 

16 

4S 

2820 

3-5457 

3010 

3.3226 

3201 

3.1240 

3395 

2.9459 

3590 

2.7852 

15 

46 

2823 

3.5418 

3013 

3.3191 

3204 

3.1209 

3398 

2.9431 

3594 

2.7827 

14 

47 

2827 

3-5379 

3016 

3.3156 

3207 

3.1178 

3401 

2.9403 

3597 

2.7801 

13 

48 

2830 

3-5339 

3019 

3.3122 

3211 

3-1146 

3404 

2.9375 

3600 

2.7776 

1 2 

49 

2833 

3.5300 

3022 

3.3087 

3214 

3-i"5 

3408 

2.9347 

3604 

2.7751 

II 

50 

2836 

3.5261 

3026 

3-3052 

3217 

3.1084 

3411 

2.9319 

3607 

2.7725 

10 

SI 

2839 

3.5222 

3029 

3-3017 

3220 

3.1053 

3414 

2.9291 

3610 

2.7700 

9 

S2 

2842 

3.5183 

3032 

3-2983 

3223 

3.1022 

3417 

2.9263 

3613 

2-7675 

8 

S3 

2845 

3-5144 

.S03S 

3.2948 

3227 

3.0991 

3421 

2-9235 

3617 

2.7650 

7 

54 

2849 

3-5105 

3038 

3-2914 

3230 

3.0961 

3424 

2.9208 

3620 

2.7625 

b 

ss 

2852 

3-5067 

3041 

3.2880 

3233 

3-0930 

3427 

2.9180 

3623 

2.7600 

5 

S6 

2855 

3-5028 

3045 

3-2845 

3236 

3.0899 

3430 

2.9152 

3627 

2-7575 

4 

S7 

28s8 

3.4989 

3048 

3.281 1 

3240 

3.0868 

3434 

2.9125 

3630 

2.7550 

3 

S8 

2861 

3-4951 

.SOS  I 

3-2777 

3243 

3-0838 

3437 

2.9097 

3633 

2.7525 

2 

59 

2864 

3.4912 

3054 

3-2743 

3246 

3.0807 

3440 

2.9070 

3636 

2.7500 

I 

GO 

2867 

3.4874 

3057 

3.2709 

3249 

3.0777 

3443 

2.9042 

3640 

2.7475 

0 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

/ 

74°    1 

73^    1 

7 

2° 

71° 

7 

0°    1  /  1 

NATURAL  TANGENTS  AND  COTANGENTS 


85 


/ 

20° 

21° 

2 

52° 

23° 

24° 

/ 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

0 

3640 

2.7475 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

60 

I 

3643 

2.7450 

3842 

2.6028 

4044 

2.4730 

4248 

2.3539 

4456 

2.2443 

59 

2 

3646 

2.7425 

3845 

2.6006 

4047 

2.4709 

4252 

2.3520 

4459 

2.2425 

58 

3 

3650 

2.7400 

3849 

2.5983 

4050 

2.4689 

4255 

2.3501 

4463 

2.240S 

57 

4 
S 

3653 

2.7376 

3852 

2.5961 

4054 

2.4668 

4258 

2.3483 

4466 

2.2390 

56 
55 

3656 

2.7351 

3855 

2.5938 

4057 

2.4648 

4262 

2.3464 

4470 

2.2373 

6 

3659 

2.7326 

3859 

2.5916 

4061 

2.4627 

4265 

2.3445 

4473 

2.2355 

54 

7 

3663 

2.7302 

3862 

2.5893 

4064 

2.4606 

4269 

2.3426 

4477 

2.2338 

53 

8 

3666 

2.7277 

.3865 

2.5871 

4067 

2.4586 

4272 

2.3407 

4480 

2.2320 

52 

9 

3669 

2.7253 

3869 

2.5848 

4071 

2.4566 

4276 

2:3388 

4484 

2.2303 

51 

10 

3673 

2.7228 

.S872 

2.5826 

4074 

2.4545 

4279 

2.3369 

4487 

2.2286 

50 

II 

367b 

2.7204 

.S875 

2.5804 

4078 

2.4525 

4283 

2.3351 

4491 

2.2268 

49 

12 

3679 

2.7179 

3879 

2.5782 

4081 

2.4504 

4286 

2.3332 

4494 

2.2251 

48 

1.3 

3(>^3 

2.7155 

3882 

2-5759 

4084 

2.4484 

4289 

2.3313 

4498 

2.2234 

47 

14 

3b86 

2.7130 

3885 

2.5737 

4088 

2.4464 

4293 

2.3294 

4501 

2.2216 

46 

IS 

3689 

2.7106 

3889 

2-5715 

4091 

2.4443 

4296 

2.3276 

4505 

2.2199 

45 

I6 

3693 

2.7082 

3892 

2.5693 

4095 

2.4423 

4300 

2.3257 

4508 

2.2182 

44 

17 

3696 

2.7058 

3895 

2.5671 

4098 

2.4403 

4303 

2.3238 

4512 

2.2165 

43 

18 

3699 

2.7034 

3899 

2.5649 

4101 

2.4383 

4307 

2.3220 

4515 

2.2148 

42 

19 

3702 

2.7009 

3902 

2.5627 

4105 

2.4362 

4310 

2.3201 

4519 

2,2130 

41 

20 

3706 

2.6985 

3906 

2.5605 

4108 

2.4342 

4314 

2.3183 

4522 

2.2113 

40 

21 

3709 

2.6961 

3909 

2.5533 

4111 

2.4322 

4317 

2.3164 

4526 

2.2096 

39 

22 

3712 

2.6937 

3912 

2.5561 

4115 

2.4302 

4320 

2.3146 

4529 

2.2079 

38 

23 

37^^ 

2.6913 

3916 

2.5539 

4118 

2.4282 

4324 

2.3127 

4533 

2.2062 

37 

24 

3719 

2.6889 

3919 

2.5517 

4122 

2.4262 

4327 

2.3109 

4536 

2.2045 

36 

25 

3722 

2.6865 

3922 

2.5495 

4125 

2.4242 

4331 

2.3090 

4540 

2.2028 

35 

2b 

3726 

2.6841 

3926 

2.5473 

4129 

2.4222 

4334 

2.3072 

4543 

2.201 1 

34 

27 

3729 

2.6818 

3929 

2.5452 

4132 

2.4202 

4338 

2.3053 

4547 

2.1994 

33 

28 

3732 

2.6794 

.3932 

2.5430 

413s 

2.4182 

4341 

2.3035 

4S50 

2.1977 

32 

29 

373^ 

2.6770 

3936 

2.5408 

4139 

2.4162 

4345 

2.3017 

4554 

2  i960 

31 

30 

3739 

2.6746 

3939 

2.5386 

4142 

2.4142 

4.348 

2.2998 

4557 

2.1943 

30 

31 

3742 

2.6723 

3942 

2.5365 

4146 

2.4122 

4352 

2.2980 

4561 

2.1926 

29 

32 

3745 

2.6699 

3946 

2.5343 

4149 

2.4102 

4355 

2.2962 

4564 

2.1909 

28 

33 

3749 

2.6675 

3949 

2.5322 

4152 

2.4083 

4359 

2.2944 

4568 

2.1892 

27 

34 

3752 

2.6652 

3953 

2.5300 

4156 

2.4063 

4362 

2.2925 

4571 

2.1876 

26 

35 

3755 

2.6628 

3956 

2.5279 

4159 

2.4043 

4365 

2.2907 

4575 

2.1859 

25 

3^ 

3759 

2.6605 

3959 

2.5257 

4163 

2.4023 

4369 

2.2889 

4578 

2.1842 

24 

37 

3762 

2.6581 

3963 

2.5236 

4166 

2.4004 

4372 

2.2871 

4582 

2.1825 

23 

3« 

3765 

2.6558 

3966 

2.5214 

4169 

2.3984 

4376 

2.2853 

4585 

2.1808 

22 

39 

3769 

2.6534 

3969 

2.5193 

4173 

2.3964 

4379 

2.2835 

4589 

2.1792 

21 

40 

3772 

2.65  II 

3973 

2.5172 

4176 

2.3945 

4383 

2.2817 

4592 

2.1775 

20 

41 

3775 

2.6488 

3976 

2.5150 

4180 

2.3925 

4386 

2.2799 

4596 

2.1758 

19 

42 

3779 

2.6464 

3979 

2.5129 

4183 

2.3906 

4390 

2.2781 

4599 

2.1742 

18 

43 

37^2 

2.6441 

3983 

2.5108 

4187 

2.3886 

4393 

2.2763 

4603 

2.1725 

17 

44 

37^5 

2.6418 

3986 

2.5086 

4190 

2.3867 

4397 

2.2745 

4607 

2.1708 

16 
15 

45 

3789 

2.6395 

3990 

2.5065 

4193 

2.3847 

4400 

2.2727 

4610 

2.1692 

4b 

3792 

2.6371 

3993 

2.5044 

4197 

2.3828 

4404 

2.2709 

4614 

2.1675 

14 

47 

3795 

2.6348 

3996 

2.5023 

4200 

2.3808 

4407 

2.2691 

4617 

2.1659 

13 

48 

3799 

2.6325 

4000 

2.5002 

4204 

2.3789 

441 1 

2.2673 

4621 

2,1642 

12 

49 

3802 

2.6302 

4003 

2.4981 

4207 

2.3770 

4414 

2.2655 

4624 

2.1625 

II 

50 

3805 

2.6279 

4006 

2.4960 

4210 

2.3750 

4417 

2.2637 

4628 

2.1609 

10 

51 

3809 

2,6256 

4010 

2.4939 

4214 

2.3731 

4421 

2.2620 

4631 

2.1592 

? 

52 

3812 

2.6233 

4013 

2.4918 

4217 

2.3712 

4424 

2.2602 

4635 

2.1576 

8 

53 

3«i5 

2.6210 

4017 

2.4897 

4221 

2.3693 

4428 

2.2584 

4638 

2.1560 

7 

54 

3«i9 

2.6187 

4020 

2.4876 

4224 

2.3673 

4431 

2.2566 

4642 

2.1543 

b 

55 

3822 

2.6165 

4023 

2.4855 

4228 

2.3654 

4435 

2.2549 

4645 

2.1527 

5 

5t> 

3«25 

2.6142 

4027 

2.4834 

4231 

2.3635 

4438 

2.2531 

4649 

2.1510 

4 

57 

3829 

2.6119 

4030 

2.4813 

4234 

2.3616 

4442 

2.2513 

4652 

2.1494 

3 

5« 

3«32 

2.6096 

4033 

2.4792 

4238 

2.3597 

4445 

2.2496 

4656 

2.1478 

2 

59 

3835 

2.6074 

4037 

2.4772 

4241 

2.3578 

4449 

2.2478 

4660 

2.1461 

I 

60 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

4663 

2.1445 

0 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

Cotg 

Tang 

/ 

69° 

68° 

67° 

66°    1 

d 

5° 

/ 

86 


TABLE    TTT 


/ 

25° 

26° 

270 

28° 

29° 

/ 

0 

2 

3 

4 

Tang  totg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

4663  2.1445 
4667  2.1429 
4670  2. 14 1 3 
4674  2.1396 
4677  2.1380 

487"7  2.0503 
4881  2.0488 
4885  2.0473 
4888  2.0458 
4892  2.0443 

5095  1.9626 
5099  1. 96 1 2 
5103  1.9598 
5106  1.9584 
5110  1.9570 

5317  1.8807 
5321  1.8794 
5325  1.8781 
5328  1.8768 
5332  1.8755 

5543  1.8040 
5547  1.8028 
5551  1. 8016 
5555  1-8003 
5558  1.7991 

60 

59 
58 
57 
56 

7 
8 

9 

4681  2.1364 
4684  2. 1 348 
4688  2.1332 
4691  2.1315 
4695  2.1299 

4895  2.0428 
4899  2.0413 
4903  2.0398 
4906  2.0383 
4910  2.0368 

51 14  1.9556 
51 17  1.9542 
5121  1.9528 
5125  1.9514 
5128  1.9500 

5336  1.8741 
5340  1.8728 

5343  1-8715 
5347  1.8702 
5351   1.8689 

5562  1.7979 
5566  1.7966 
5570  1.7954 
5574  1.7942 
5577  1-7930 

55 
54 
53 
52 
51 

10 

II 

12 

13 

14 

4699  2.1283 
4702  2.1267 
4706  2. 1 25 1 
4709  2.1235 
4713  2.1219 

4913  2.0353 
4917  2.0338 
4921  2.0323 
4924  2.0308 
4928  2.0293 

5132  1.9486 
5136  1.9472 
5139  1.9458 
5143  1.9444 
5147  1.9430 

5354  1.8676 
5358  1.8663 
5362  1.8650 
5366  1.8637 
5369  1.8624 

5581  1.7917 
5585  1.7905 
5589  1-7893 
5593  1.7881 
5596  1.7868 

50 

49 
48 

11 

17 
19 

4716  2.1203 
4720  2.1 187 
4723  2.1 171 
4727  2.1155 
4731  2.1 139 

4931  2.0278 
4935  2.0263 
4939  2.0248 
4942  2.0233 
4946  2.0219 

5150  1.9416 
5154  1.9402 
5158  1.9388 

5161  1.9375 
5165  1.9361 

5373  1861 1 
5377  1-8598 
5381  1.8585 
5384  1.8572 
5388  1.8559 

5600  1.7856 
5604  1.7844 
5608  1.7832 
5612  1.7820 
5616  1.7808 

45 
44 
43 
42 
41 

20 

21 

22 

23 

24 

4734  2. 1 1 23 
4738  2.1 107 
4741  2.1092 
4745  2.1076 
4748  2.1060 

4950  2.0204 
4953  2.0189 
4957  2.0174 
4960  2.0160 
4964  2.0145 

5169  1.9347 

5172  1.9333 
5176  1.9319 
5180  1.9306 
5184  1.9292 

5392  1.8546 
5396  1.8533 
5399  1.8520 
5403  1.8507 
5407  1.8495 

5619  1.7796 

5623  1.7783 
5627  1.7771 

5631  1.7759 
5635  1.7747 

40 

38 
37 
36 

27 
28 
29 

4752  2.1044 
4755  2.1028 
4759  2.1013 
4763  2.0997 
4766  2.0981 

4968  2.0130 
4971  2.0115 
4975  2.0101 
4979  2.0086 
4982  2.0072 

5187  1.9278 
5191  1.9265 

5^95  1-9251 
5198  1.9237 
5202  1.9223 

541 1   1.8482 
5415  1.8469 
5418  1.8456 
5422  1.8443 
5426  1.8430 

5639  1.7735 
5642  1.7723 
5646  1.7711 
5650  1.7699 
5654  1.7687 

35 
34 
33 
32 
31 

30 

31 
32 
33 

34 

4770  2.0965 
4/  73     2.0950 
4777  2.0934 
4780  2.0918 
4784  2.0903 

4986  2.0057 
4989  2.0042 
4993  2.0028 
4997  2.0013 
5000   1.9999 

5206  1.9210 
5209  1.9196 
5213  1.9183 
5217  1.9169 
5220  1.9155 

5430  1. 84 1 8 

5433  1.8405 
5437  1.8392 
5441   1.8379 
5445   1.8367 

5658  1.7675 
5662  1.7663 

5665  1.7651 
5669  1.7639 
5673  1.7627 

30 

29 
28 
27 
26 

39 

4788  2.0887 
4791  2.0872 
4795  2.0856 
4798  2.0840 
4802  2.0825 

5004   1.9984 
5008  1.9970 

50"   1-9955 
5015   1. 9941 
5019  1.9926 

5224  1.9142 
5228  1.9128 
5232  1.9115 
5235  1.9101 
5239  1.9088 

5448  1.8354 
5452  1.8341 
5456  1.8329 
5460  1. 83 1 6 
5464  1.8303 

5677  1.7615 
5681   1.7603 
5685  1.7591 
5688  1.7579 
5692  1.7567 

25 

24 

23 
22 

21 

40 

41 
42 

43 
44 

4806  2.0809 
4809  2.0794 
4813  2.0778 
4816  2.0763 
4820  2.0748 

5022  1. 9912 
5026  1.9897 
5029  1.9883 
5033  1.9868 
5037  1.9854 

5243  1.9074 
5426  1. 9061 
5250  1.9047 
5254  1.9034 
5258  1.9020 

5467   1. 8291 
5471   1.8278 
5475  1-8265 
5479  1.8253 
5482  1.8240 

5696  1.7556 
5700  1.7544 
5704  1.7532 
5708  1.7520 
5712  1.7508 

20 

19 
18 

17 
16 

45 
46 

47 
48 

49 

4823  2.0732 
4827  2.0717 
4831  2.0701 
4834  2.0686 
4838  2.0671 

5040  1 .9840 
5044  J. 9825 
5048  1. 98 II 
5051   1.9797 
5055  1.9782 

5261   1.9007 
5265  1.8993 
5269  1.8980 
5272  1.8967 
5276  1.8953 

5486  1.8228 
5490  1.8215 
5494  1.8202 
5498  1. 8 1 90 
5501   1.8177 

5715  1.7496 

5719  1.7485 
5723  1.7473 
5727  1. 7461 
5731   1.7449 

15 
14 
13 
12 
II 

50 

51 

52 

53 
54 

4841  2.0655 
4845  2.0640 
4849  2.0625 
4852  2.0609 
4856  2.0594 

5059  1.9768 
5062  1.9754 
5066  1.9740 
5070  1.9725 
5073  1.9711 

5280  1.8940 
5284  1.8927 
5287  1.8913 
5291   1.8900 
5295  1.8887 

5505  1-8165 
5509  1.8152 
5513  1.8140 
5517  1.8127 
5520  1.8115 

5735  1-7437 
5739  1.7426 

5743  1-7414 
5746  1.7402 
5750  1.7391 

10 

9 
8 

7 
6 

59 

4859  2.0579 
4863  2.0564 
4867  2.0549 
4870  2.0533 
4874  2.0518 

5077  1.9697 
5081   1.9683 
5084  1.9669 
5088  1.9654 
5092  1.9640 

5298  1.8873 
5302  1.8860 
5306  1.8847 
5310  1.8834 
5313  1.8820 

5524  1.8103 
5528  1.8090 
5532  1.8078 
5535  1.8065 
5539  1.8053 

5754  1-7379 
5758  1.7367 
5762  1.7355 
5766  1.7344 
5770  1.7332 

5 
4 
3 
2 
I 

60 

4877  2.0503 

5095  1.9626 

5317  1.8807 

5543  1.8040 

5774  1-7321 

0 

Cotff  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

/ 

64° 

63° 

620 

61° 

60°     /  1 

NATURAL  TANGENTS  AND  COTANGENTS 

87 

/ 

30° 

31° 

32° 

38° 

34°     /  1 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

0 

I 

2 

3 
4 

5 
6 

7 
8 

9 

5774  1.7321 
5777  1-7309 
5781   1.7297 
5785  1.7286 
5789  1.7274 

6009  1.6643 
6013  1.6632 
6017  1.6621 
6020  1. 66 10 
6024  1.6599 

6249  1.6003 
6253  1.5993 
6257  1.5983 
6261  1.5972 
6265  1.5962 

6494  1.5399 
6498  1.5389 
6502  1.5379 
6506  1.5369 
6511   1-5359 

6745  1.4826 
6749  1.4816 
6754  1.4807 
6758  1.4798 
6762  1.4788 

60 

59 
58 
57 
56 

5793  1-7262 
5797  1-7251 
5801   1.7239 
5805   1.7228 
5808  1.7216 

6028   1.6588 
6032   1.6577 
6036  1.6566 
6040  1.6555 
6044  1.6545 

6269  1.5952 
6273  1.5941 
6277  1.5931 
6281  1.5921 
6285  1. 59 1 1 

6515  1-5350 
6519  1.5340 

6523  1-5330 
6527  1-5320 
6531   1-53" 

6766  1.4779 
6771  1.4770 
6775  1.4761 
6779  1.4751 
6783  1.4742 

55 
54 
53 
52 
51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 
40 
39 
38 
37 
36 

10 

II 

12 

13 

5812  1.7205 
5816  1.7193 
5820  1.7182 
5824  1.7170 
5828  1.7159 

6048  1.6534 
6052   1.6523 
6056  1.65 1 2 
6060   1. 6501 
6064  1.6490 

6289  1.5900 
6293  1.5890 
6297  1.5880 
6301  1.5869 
6305  1.5859 

6536  1.5301 
6540  1.5291 
6544  1.5282 
6548  1.5272 
6552  1.5262 

6787  1.4733 
6792  1.4724 
6796  1.4715 
6800  1.4705 
6805  1.4696 

15 
i6 

17 
i8 

19 

5832  1. 7147 
5836  1,7136 
5840  1.7124 
5844  1.7113 
5847  1. 7 102 

6068   1.6479 
6072  1.6469 
6076  1.6458 
6080   1.6447 
6084  1.6436 

6310  1.5849 
6314  1.5839 
6318  1.5829 
6322  •  1.5818 
6326  1.5808 

6556  1.5253 
6560  1.5243 

6565  1.5233 
6569  1.5224 
6573  1.5214 

6809  1.4687 
6813  1.4678 
6817  1.4669 
6822  1.4659 
6826  1.4650 

20 

21 

22 

23 
24 

5851   1.7090 

5S55   1-7079 
5859  1.7067 
5863  1.7056 
5S67  1.7045 

6088  1.6426 
6092  1. 641 5 
6096  1.6404 
6100  1.6393 
6104  1.6383 

6330  1.5798 
6334  1-5788 
6338  1.5778 
6342  1.5768 
6346  1.5757 

6577  1.5204 

6581   1.5195 
6585  1.5185 

6590  1.5 1 75 
6594  1.5 166 

6830  1. 464 1 
6834  1.4632 
6839  1.4623 
6843  1. 46 14 
6847  1.4605 

25 
26 
27 
28 
29 

5871   1.7033 
5875   1.7022 
5879  1.7011 
5883  1.6999 
5887  1.6988 

6108   1.6372 
6112   1.6361 
61 16   1.6351 
6120  1.6340 
6124  1.6329 

6350  1.5747 
6354  1.5737 
6358  1.5727 

6363  1-5717 
6367  1-5707 

6598  1.5156 
6602  1.5 147 
6606  1.5 137 
6610  1. 5127 
6615  1.5118 

6851  1.4596 
6856  1.4586 
6860  1.4577 
6864  1.4568 
6869  1.4559 

35 
34 
33 
32 
31 

30 

31 

32 
33 
34 

5890  1.6977 
5894  1.6965 
5898  1.6954 
5902  1.6943 
5906  1.6932 

6128  1.6319 
6132  1.6308 
6136  1.6297 
6140  1.6287 
6144  1.6276 

6371  1.5697 
6375  1-5687 
6379  1-5677 
6383  1.5667 
6387  1.5657 

6619  1.5108, 
6623  1.5099 
6627  1.5089 
6631   1.5080 
6636  1.5070 

6873  1.4550 
6877  1.4541 
6881  1.4532 
6886  1.4523 
6890  1.45 14 

30 

29 

28 

27 
26 

35 
36 
37 
38 
39 

5910  1.6920 
5914  1.6909 
5918  1.6898 
S922  1.6887 
5926  1.6875 

6148   1.6265 
6152  1.6255 
6156  1.62,1/1 
6160  1.6234 
6164   1.6223 

6391  1.5647 

6395  1-5637 
6399  1-5627 
6403  1. 56 1 7 
6408  1.5607 

6640  1. 506 1 
6644  1.505 1 
6648  1.5042 
6652  1.5032 
6657  1.5023 

6894  1.4505 
6899  1.4496 
6903  1.4487 
6907  1.4478 
691 1   1.4469 

25 
24 
23 
22 
21 

40 

41 

42 

43 

44 

5930  1.6864 
5934  1.6853 
5938  1.6842 
5942  1.6831 
5945   1.6820 

6168  1.6212 
6172   1.6202 
6176  1.6191 
6i8o  1.6181 
6184  1. 6170 

6412  1.5597 
6416  1.5587 
6420  1.5577 
6424  1.5567 
6428  1.5557 

6661   1.5013 
6665  1.5004 
6669  1.4994 
6673  1.4985 
6678  1.4975 

6916  1.4460 
6920  1.445 1 
6924  1.4442 
6929  1.4433 
6933  1-4424 

20 

19 
18 

17 
16 

45 
46 

47 

48 

49 
50 

51 

52 
53 
54 

5949  1.6808 
5953  1-6797 
5957  1-6786 
5961   1.6775 
5965   1.6764 

6188   1.6160 
6192  1.6149 
6196  1.6139 
6200  1. 6 1 28 
6204  1. 61 18 

6432  1.5547 

6436  1.5537 
6440  1.5527 

6445  1-55 1 7 
6449  1-5507 

6682  1.4966 
6686  1.4957 
6690  1.4947 
6694  1.4938 
6699  1.4928 

6937  1.4415 
6942  1.4406 
6946  1.4397 
6950  1.4388 
6954  1-4379 

15 
14 
13 
12 

5969  1.6753 
5973  1-6742 
5977  1-6731 
5981   1.6720 
5985   1.6709 

6208  1.6107 
6212  1.6097 
6216  1.6087 
6220  1.6076 
6224  1.6066 

6453  1-5497 
6457  1.5487 
6461   1.5477 
6465   1.5468 
6469  1-5458 

6703  1.4919 
6707   1. 49 10 
6711   1.4900 
6716  1. 489 1 
6720  1.4882 

6959  1-4370 
6963  1.4361 

6967  1-4352 
6972  1.4344 

6976  1-4335 

10 

9 
8 

7 
6 

55 
56 

11 

59 

5989  1.6698 
5993  1.6687 
5997   1-6676 
6001   1.6665 
6005   1.6654 

6228  1.6055 
6233  1.6045 
6237   1-6034 
6241   1.6024 
6245   1. 6014 

6473  1-5448 
6478  1.5438 
6482  1.5428 
6486  1.5418 
6490  1.5408 

6724  1.4872 
6728  1.4863 
6732  1.4854 
6737  1.4844 
6741   1.4835 

6980  1.4326 
6985   1.43 1 7 
6989  1.4308 

6993  1-4299 
6998  1.4290 

5 
4 
3 
2 

GO 

6009   1.6643 

6249   1.6003 

6494   1-5399 

6745  1.4826 

7002  1. 4281 

0 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

/ 

51)° 

5S° 

57° 

56° 

55° 

/ 

88 


TABLE  III 


/ 

35° 

36° 

37° 

38° 

31)° 

/ 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

0 

I 

2 

3 

4 

7002  1. 428 1 
7006  1.4273 
70 II  1.4264 
7015  1.4255 
7019  1.4246 

7265  1.3764 
7270  1.3755 
7274  1.3747 

7279  1-3739 
7283  1.3730 

7536  1.3270 
7540  1.3262 
7545  1.3254 
7549  1.3246 
7554  1.3238 

7813  1.2799 
7818  1.2792 
7822  1.2784 
7827  1.2776 
7832  1.2769 

S098  1.2349 
8103  1.2342 
8107  1.2334 
81 12  1.2327 
8117  1.2320 

60 

59 
58 

I 

7 
8 

9 

7024  1.4237 
7028  1.4229 
7032  1.4220 
7037  1.4211 
7041  1.4202 

7288  1.3722 
7292  1.3713 
7297  1.3705 
7301   1.3697 
7306  1.3688 

7558  1.3230 
7563  1.3222 
7568  1.3214 
7572  1.3206 
7577  1.3198 

7836  1.2761 

7841  1.2753 
7846  1.2746 
7850  1.2738 
7855  1-2731 

8122  1. 2312 
8127  1.2305 
8132  1.2298 
8136  1.2290 
8141  1.2283 

55 
54 
53 
52 
51 

10 

II 

12 

13 
14 

7046  1. 41 93 
7050  1. 4 1 85 
7054  1.4176 
7059  1.4167 
7063  1. 41 58 

7310  1.3680 
7314  1.3672 
7319  1.3663 

7323  1.3655 
7328  1.3647 

7581   1.3190 
7586  1.3182 
7590  1.3175 
7595  1.3167 
7600  1. 3 1 59 

7860  1.2723 
7865  1.2715 
7869  1.2708 
7874  1.2700 
7879  1.2693 

8146  1.2276 
8151  1.2268 
8156  1. 2261 
8i6i   1.2254 
8165  1.2247 

60 

49 
48 

47 
46 

19 

7067  1.4150 
7072  1.4141 
7076  1.4132 
7080  1. 4 1 24 
7085  1.4115 

7332  1.3638 
7337  1.3630 
7341   1.3622 
7346  1.3613 
7350  1.3605 

7604  1.3151 
7609  1.3143 
7613  1.3135 
7618  1.3127 
7623  1.3119 

7883  1.2685 
7888  1.2677 
7893  1.2670 
7898  1.2662 
7902  1.2655 

8170  1.2239 
8175  1.2232 
8180  1.2225 
8185   1.2218 
8190  1. 2210 

45 
44 
43 
42 
41 

20 

21 
22 
23 

24 

7089  1. 4 1 06 
7094  1.4097 
7098  1.4089 
7102  1.4080 
7107  1.4071 

7355  1-3597 
7359  1.3588 
7364  1.3580 
7368  1.3572 
7373  1-3564 

7627  1.3111 
7632  1. 3103 

7636  1.3095 
7641  1.3087 
7646  1.3079 

7907  1.2647 
7912  1.2640 
7916  1.2632 
7921  1.2624 
7926  1.2617 

8195  1.2203 
8199  1. 2196 
8204  1. 21 89 
8209  1.2181 
8214  1.2174 

40 

39 
38 
27 
36 

25 
26 
27 
28 
29 

71 II  1.4063 
71 15  1.4054 
7120  1.4045 
7124  1.4037 
7129  1.4028 

7377  1.3555 
7382  1.3547 

7386  1.3539 
7391   1.3531 
7395  1.3522 

7650  1.3072 
7655  1.3064 
7659  1.3056 
7664  1.3048 
7669  1.3040 

7931  1.2609 
7935  1.2602 
7940  1.2594 
7945  1.2587 
7950  1.2579 

8219  1. 2167 
8224  1.2 1 60 
8229  1.2153 
8234  1.2145 
8238  1.2138 

35 
34 

32 
31 

30 

31 

32 

zz 

34 

7133  1.4019 
7137  1.4011 
7142  1.4002 
7146  1.3994 
7151  1.3985 

7400  1.35 14 
7404  1.3506 
7409  1.3498 
7413  1.3490 
7418  1. 3481 

7673  1.3032 
7678  1.3024 
7683  1.3017 
7687  1.3009 
7692  1.3001 

7954  1.2572 

7959  1.2564 
7964  1.2557 
7969  1.2549 
7973  1.2542 

8243  1.2131 
8248  1. 21 24 
8253  1.2117 
8258  1.2109 
8263  1.2102 

30 

29 
28 

27 
26 

37 
38 
39 

7155  1-3976 
7159  1.3968 
7164  1.3959 
7168  1.3951 
7173  1.3942 

7422  1.3473 
7427  1.3465 
7431   1.3457 
7436  1.3449 
7440  1.3440 

7696  1.2993 
7701   1.2985 
7706  1.2977 
7710  1.2970 
7715  1.2962 

7978  1.2534 
7983  1.2527 
7988  1.2519 
7992  1.25 1 2 
7997  1.2504 

8268  1.2095 
8273  1.2088 
8278  1.2081 
8283  1.2074 
8287  1.2066 

25 
24 
23 
22 
21 

40 

41 

42 

43 
44 

7177  1.3934 
7181  1.3925 
7186  1.3916 
7190  1.3908 
7195  1-3899 

7445   1.3432 
7449   1.3424 
7454   1.3416 
7458  1.3408 
7463   1.3400 

7720  1.2954 
7724  1.2946 
7729  1.2938 

7734  1.2931 
7738  1.2923 

8002  1.2497 
8007  1.2489 
8012  1.2482 
8016  1.2475 
8021   1.2467 

8292  1.2059 
8297  1.2052 
8302  1.2045 
8307  1.2038 
8312  1.203 1 

20 

»9 

18 

17 
16 

45 
46 

47 
48 

49 

7199  1.3891 
7203  1.3882 
7208  1.3874 
7212  1.3865 
7217  1.3857 

7467   1.3392 
7472  1.3384 
7476  1.3375 
7481   1.3367 

7485   1.3359 

7743  1.2915 

7747  1.2907 
7752  1.2900 
7757  1.2892 
7761   1.2884 

8026  1.2460 
8031   1.2452 
8035  1.2445 
8040  1.2437 
8045   1.2430 

8317  1.2024 
8322  1. 2017 
8327  1.2009 
8332  1.2002 
8337  1.1995 

15 
14 
13 
12 
II 

50 

51 

52 
53 
54 

7221  1.3848 
7226  1.3840 
7230  1.3831 
7234  1.3823 
7239  1.3814 

7490  1.335 1 
7495  1-3343 
7499  1.3335 
7504  1.3327 
7508  1.3319 

7766  1.2876 
7771   1.2869 
7775  1.2861 
7780  1.2853 
7785  1.2846 

8050  1.2423 
8055  1.2415 
8059  1.2408 
8064  1. 240 1 
8069  1.2393 

8342  1. 1 988 
8346  1.1981 
8351   1. 1974 
8356  1. 1967 
8361   1. 1960 

10 

9 
8 

7 
6 

55 
56 

59 

7243  1.3806 
7248  1.3798 
7252  1.3789 
7257  1.3781 
7261  1.3772 

7513  i-2>2,^i 
7517   1.3303 
7522  1.3295 
7526  1.3287 
7531   1.3278 

7789  1.2838 
7794  1.2830 
7799  1.2822 
7803  1.28 1 5 
7808  1.2807 

8074  1.2386 
8079  1.2378 
8083  1. 237 1 
8088  1.2364 
8093  1.2356 

8366  1. 1953 
8371   1. 1946 
8376  1. 1939 
8381   1. 1932 
8386  1. 1925 

5 
4 
3 
2 

60 

7265  1.3764 

7536  1.3270 

7813  1.2799 

8098  1.2349 

8391   1.1918 

0 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

_ 

/ 

54° 

53° 

52° 

51° 

50° 

zl 

NATURAL  TANGENTS  AND  COTANGENTS 


/ 

40° 

41° 

42o 

43° 

44° 

/ 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

Tang  Cotg 

60 

59 

58 

11 

0 

I 

2 

3 
4 

8391  1.1918 
8396  1.1910 
8401  1. 1903 
8406  1. 1 896 
8411  1. 1889 

8693  1.1504 
8698  1.1497 
8703  1.1490 
8708  1.1483 
8713  1.1477 

9004  I.I  106 
9009  I.I  100 
9015  1.1093 
9020  1.1087 
9025  1.1080 

9325  1.0724 
9331  1.0717 
9336  1.0711 
9341  1.0705 
9347  1.0699 

9657  1-0355 
9663  1.0349 
9668  1.0343 

9674  1-0337 
9679  1.0331 

1, 

9 

8416  1. 1882 
8421  I. 1875 
8426  1. 1 868 
8431   1.1861 
8436   1. 1 854 

8718  1.1470 
8724  1.1463 
8729  1. 1456 
8734  1.1450 
8739  1. 1443 

9030  1.1074 
9036  1.1067 
9041  1.1061 
9046  1. 1 054 
9052  1. 1048 

9352  1.0692 
9358  1.0686 
9363  1.0680 
9369  1.0674 
9374  1.0668 

9685  1.0325 
9691  1.0319 
9696  1. 03 1 3 
9702  1.0307 
9708  1. 030 1 

55 
54 
53 
52 
51 
50 
49 
48 
47 
46 

10 

II 

12 

13 
14 

8441   1. 1 847 
8446  1. 1 840 

8451   1.1833 
8456  1. 1826 
8461   1.1819 

8744  1.1436 
8749  1.1430 
8754  1.1423 
8759  1.1416 
8765  1.1410 

9057  1.1041 
9062  1.1035 
9067  1.1028 
9073  1.1022 
9078  1. 1016 

9380  1.0661 
9385  1.0655 
9391  1.0649 
9396  1.0643 
9402  1.0637 

9713  1.0295 
9719  1.0289 
9725  1.0283 
9730  1.0277 
9736  1.0271 

15 

16 

17 

18 

19 

8466  1.1812 
8471   1. 1806 
8476   I.I  799 
8481   1. 1792 
8486  1.17^5 

8770  1.1403 
8775  1-1396 
8780  1.1389 

8785  1.1383 
8790  1.1376 

9083  1.1009 
9089  1. 1 003 
9094  1.0996 
9099  1.0990 
9105  1.0983 

9407  1.0630 
9413  1.0624 
9418  1.0618 
9424  1. 06 1 2 
9429  1.0606 

9742  1.0265 
9747  1-0259 
9753  1-0253 
9759  1.0247 
9764  1.0241 

45 
44 
43 
42 

41 

20 

21 

22 

23 

24 

8491   I. 1778 
8496  1.1771 
8501   1.1764 
8506  1. 1757 
8511   1. 1750 

8796  1. 1 369 
8801  1. 1363 
8806  1.1356 
8811  1.1349 
8816  1.1343 

9110  1.0977 
9115  1.0971 
91 21  1.0964 
9126  1.0958 
9131  1.0951 

9435   1-0599 
9440  1.0593 
9446  1.0587 
9451   1.0581 
9457   1-0575 

9770  1.0235 
9776  1.0230 
9781   1.0224 
9787  1.0218 
9793  1. 02 1 2 

40 

39 

3^ 

11 

25 
26 
27 
28 
29 

8516  1. 1743 
8521   1. 1736 
8526  1. 1729 
8531   1. 1722 
8536   1.1715 

8821  1. 1336 
8827  1.1329 
8832  1.1323 
8837  1.1316 
8842  1.1310 

9137  1.0945 
9142  1.0939 
9147  1.0932 
9153  1.0926 
9158  1.0919 

9462  1.0569 
9468  1.0562 
9473  1-0556 
9479  1-0550 
9484  1.0544 

9798  1.0206 
9804  1.0200 
9810  1.0194 
9816  1.0188 
9821   1.0182 

35 
34 
33 
32 
31 

30 

31 
32 

33 
34 

8541   I. 1708 
8546  I.I  702 

8551   1-1695 
8556  1. 1688 
8561   1.1681 

8847  1.1303 
8852  1.1296 
8858  1.1290 
8863  1.1283 
8868   1.1276 

9163  1.0913 
9169  1.0907 
9174  1.0900 
9179  1.0894 
9185  1.0888 

9490  1.0538 
9495   1-0532 
9501   1.0526 
9506  1.0519 
9512   1.0513 

9827  1.0176 
9833  1.0170 
9838  1. 01 64 
9844  1. 01 58 
9850  1.0152 

30 

29 
28 
27 
26 

37 
38 
39 

8566  1.1674 
8571   1.1667 
8576  1.1660 
8581   I. 1653 
8586  1. 1 647 

8873  1.1270 
8878  1.1263 
8884   1.1257 
8889   1. 1250 
8894  1.1243 

9190  1.0881 
9195  1.0875 
9201  1.0869 
9206  1.0862 
9212  1.0856 

9517   1.0507 
9523  1.0501 
9528   1.0495 
9534  1.0489 
9540  1.0483 

9856  1.0147 
9861   1.0141 
9867  1.0135 
9873  1.0129 
9879  1.0123 

25 
24 

23 
22 
21 

40 

41 
42 
43 
44 

i 

49 

8591   I. 1640 
8596  I. 1633 
8601   1. 1 626 
8606   1. 1619 
8611   1. 1612 

8899   1-1237 
8904  I.I  230 
8910  1.1224 
8915   1.1217 
8920  1.1211 

9217  1.0850 
9222  1.0843 
9228  1.0837 
9233  1. 083 1 
9239  1.0824 

9545   1-0477 
9551   1.0470 
9556   1.0464 
9562  1.0458 
9567   1.0452 

9884  1.0117 

9890   l.OIIl 
9896  1.0105 
9902  1.0099 
9907  1.0094 

20 

19 
18 

17 
16 

8617  1. 1606 
8622   1. 1599 
8627   1. 1592 
8632  1.1585 
8637   1.1578 

8925   1.1204 
8931   1.1197 
8936  1.1191 
8941   I.I  184 
8946  1.1178 

9244  1. 08 1 8 
9249  1. 08 1 2 
9255   1.0805 
9260  1.0799 
9266   1.0793 

9573  1.0446 
9578  1.0440 
9584  1.0434 
9590  1.0428 
9595  1.0422 

9913  1.0088 
9919  1.0082 
9925   1.0076 
9930  1.0070 
9936  1.0064 

15 
14 
13 
12 
II 

50 

51 

52 
53 
54 

8642  1.1571 
8647   1. 1 565 
8652  1.1558 
8657   1.1551 
8662   1.1544 

8952  1.1171 
8957   1.1165 
8962  1.1158 
8967   1.1152 
8972   1.1145 

9271   1.0786 
9276  1.0780 
9282  1.0774 
9287  1.0768 
9293  1.0761 

9601   1.0416 
9606  1. 04 10 
9612  1.0404 
9618  1.0398 
9623  1.0392 

9942  1.0058 
9948  1.0052 
9954  1.0047 
9959  I -004 1 
9965   1.0035 

10 

i 

7 
6 

57 
58 
59 

8667   1. 1538 
8672  1.1531 
8678  1.1524 
8683   1.1517 
8688  1. 1510 

8978  I.I  139 
8983   1.1132 
8988  1.1126 
8994  1.1119 
8999  I-III3 

9298  1.0755 

9303   1-0749 
9309  1.0742 
9314  1.0736 
9320  1.0730 

9629  1.0385 

9634  1-0379 
9640  1.0373 
9646  1.0367 
9651   1.0361 

9971   1.0029 
9977  1.0023 
9983  1.0017 
9988  1. 00 1 2 
9994  1.OC06 

5 

4 
3 
2 

I 

(>0 

8693  1.1504 

9004  I.I  106 

9325   1.0724 

9657  1-0355 

1000  1. 0000 

0 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

Cotg  Tang 

/ 

490 

48° 

47° 

46° 

45° 

/ 

TABLE   IV 

* 

SQUARES  OF  NUMBERS 

No. 

Square. 

No. 

Square. 

No. 

Square. 

No. 

Square. 

No. 

Square. 

0 

I 

O 

20 
21 

400 

40 

41 

1600 

m 
61 

3600 

80 
81 

64CX) 

I 

441 

1681 

3721 

6561 

2 

4 

22 

484 

42 

1764 

62 

3844 

82 

6724 

3 

9 

23 

529 

43 

1849 

63 

3969 

83 

6889 

4 

i6 

24 

576 

44 

1936 

64 

4096 

84 

7056 

5 

25 

25 

625 

45 

2025 

65 

4225 

85 

7225 

6 

36 

26 

676 

46 

2116 

66 

4356 

86 

7396 

7 

49 

27 

729 

47 

2209 

67 

4489 

87 

7569 

8 

64 

28 

784 

48 

2304 

68 

4624 

88 

7744 

9 
10 

1 1 

81 

29 
30 

31 

841 

49 
50 

51 

2401 

69 
70 

71 

4761 

89 
90 

91 

7921 
8100 

100 

900 

2500 

4900 

121 

961 

2601 

5041 

8281 

12 

144 

32 

1024 

52 

2704 

72 

5184 

92 

8464 

'3 

169 

33 

1089 

53 

2809 

73 

5329 

93 

8649 

14 

196 

34 

1 1 56 

54 

2916 

74 

5476 

94 

8836 

'5 

22s 

35 

1225 

55 

3025 

75 

5625 

95 

9025 

i6 

256 

36 

1296 

56 

3136 

76 

5776 

96 

9216 

17 

289 

37 

1369 

57' 

3249 

77 

5929 

97 

9409 

i8 

324 

38 

1444 

58' 

3364 

78 

6084 

98 

9604 

«9 
•20 

361 
400 

39 
40 

1521 

59 
GO 

3481 

79 
80 

6241 

99 
100 

9801 

1600 

3600 

6400 

lOOOO 

91 


92 


TABLE  IV 


00 

!♦♦ 

244 

$♦♦ 

4^# 

&♦♦ 

6#^ 

T4^ 

§♦♦ 

9^^ 

u 

00 

Diff' 

I 

100 

400 

900 

1600 

2500 

3600 

4900 

6400 

8100 

OI 

02 
03 

102 
104 
106 

404 
408 
412 

906 
912 
918 

1608 
1616 
1624 

2510 
2520 
2530 

3612 

4914 
4928 
4942 

6416 
6432 
6448 

8118 
8136 
8154 

01 

04 
09 

3 

5 
7 

04 

108 
no 
112 

416 

420 
424 

924 
930 
936 

1632 
1640 
1648 

2540 

2550 
2560 

3648 
3660 
3672 

4956 

6464 
6480 
6496 

8172 
8190 
8208 

16 

25 

36 

9 
II 

13 

11 

09 

114 
116 
118 

428 
432 
436 

04.2 
948 

954 

1656 
1664 
1672 

2570 
2580 
2590 

3696 

3708 

4998 
5012 
5026 

6512 
6528 
6544 

8226 

8244 
8262 

1'. 

IS 
17 
19* 

10 

121 

441 

961 

1681 

2601 

3721 

5041 

6561 

8281 

00 

21 

II 

12 
13 

123 

125 

127 

445 
449 
453 

967 
973 
979 

1689 
1697 
1705 

2611 
2621 
2631 

3733 
3745 
3757 

5055 
5069 
5083 

6577 

8299 
8317 
8335 

21 

44 
69 

23 
25 
27 

14 

129 
132 
134 

til 

466 

985 
992 
998 

1713 

1722 
1730 

2641 
2652 
2662 

3769 
3782 

3794 

5097 
5112 
5126 

6625 
6642 
6658 

8353 
8372 
8390 

96 
5^ 

29* 

31 

33 

19 

136 
139 
141 

470 

475 
479 

1004 
ion 
1017 

1738 

1747 
1755 

2672 
2683 
2693 

3806 
3819 
3^3^ 

5140 
5155 
51^9 

6691 
6707 

8408 
8427 
8445 

89 
6i 

35* 

37 

39* 

20 

144 

484 

1024 

1764 

2704 

3844 

5184 

6724 

8464 

00 

4» 

21 

22 
23 

146 
148 
151 

488 
492 
407 

1030 
1036 
1043 

1772 
17S0 
1789 

2714 

2724 
2735 

3881 

5198 
5212 
5227 

6756 
^773 

8482 
8500 
8519 

41 

84 

29 

43 

45* 

47 

24 

153 

lit 

501 
506 
510 

1049 
1056 
1062 

1797 
1806 
1814 

2745 
2766 

3893 
3906 
3918 

5241 
5256 
5270 

6789 
6806 
6822 

8537 
8556 
8574 

76 
25 
76 

49* 
5» 

53* 

11 

29 

161 

166 

515 
519 
524 

1069 

1075 
1082 

1823 
1831 
1840 

2777 
2787 
2798 

3931 
3943 
3956 

5285 
5299 
53H 

6839 

6855 
6872 

8630 

29 
84 
41 

55 
51* 

59* 

30 

169 

529 

1089 

1849 

2809 

3969 

5329 

6889 

8649 

00 

61 

31 
32 
33 

171 

III 

542 

1095 
1102 
1 108 

1857 
1866 
1874 

2819 
2830 
2840 

3981 

3994 
4006 

5343 
5358 
5372 

6905 
6922 
6938 

8667 
8686 
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00 

MATHEMATICAL   SERIES 

While  this  series  has  been  planned  to  meet  the 
needs  of  the  student  who  is  preparing  for  engineer- 
ing work,  it  is  hoped  that  it  will  serve  equally  well 
the  purposes  of  those  schools  where  mathematics  is 
taken  as  an  element  in  a  liberal  education.  In  order 
that  the  applications  introduced  may  be  of  such  char- 
acter as  to  interest  the  general  student  and  to  train 
the  prospective  engineer  in  the  kind  of  work  which 
he  is  most  likely  to  meet,  it  has  been  the  policy  of  the 
editors  to  select  as  joint  authors  of  each  text,  a 
mathematician  and  a  trained  engineer  or  physicist. 

The  problems  as  well  as  the  applications  intro- 
duced in  the  text  are  of  such  a  character  as  to  draw 
upon  the  student's  general  information  which  will 
be  of  use  to  him  later  in  the  application  of  mathe- 
matics. Without  sacrificing  the  value  of  mathe- 
matical study  as  a  discipline,  it  is  the  purpose  of  the 
series  so  to  correlate  the  mathematics  with  the  phy- 
sical applications  as  to  stimulate  the  interest  and 
train  the  student  to  use  his  mathematics  as  a  means 
of  investigation  and  stating  the  laws  of  physical 
phenomena. 

The  following  texts  have  appeared : 

I.  Calculus. 

By  E.  J.  TowNSEND,  Professor  of  Mathematics  in  the 
University  of  Illinois,  and  G.  A.  Goodenough,  Professor  of 
Mechanical  Engineering,  University  of  Illinois.     $2.50. 

II.  College  Algebra. 

By  H.  L.  RiETZ,  Assistant  Professor  of  Mathematics  in 
the  University  of  Illinois,  and  Dr.  A.  R.  Crathorne,  Asso- 
ciate in  Mathematics  in  the  University  of  Illinois.     $1.40. 

III.  Trigonometry. 

By  A.  G.  Hall,  Professor  of  Mathematics  in  the  Uni- 
versity of  Michigan,  and  F.  G.  Frink,  Professor  of  Railway 
Engineering  in  the  University  of  Oregon.     $1.25. 

HENRY      HOLT      AND      COMPANY 

NEW  YORK  CHICAGO 


ENGINEERING  BOOKS 

Hoskins's  Hydraulics. 

By  L.  M.  HosKiNS,  Professor  in  Leland  Stanford  Uni- 
versity.   8vo.    271  pp.    $2.50. 

A  comprehensive  text-book,  intended  for  the 
fundamental  course  in  the  subject  usually  offered 
in  schools  of  engineering,  but  somewhat  more  com- 
pact in  treatment  than  the  ordinary  treatise  now 
available. 

Russell's  Text-book  on  Hydraulics. 

By  George  E.  Russell,  Assistant  Professor  of  Civil  En- 
gineering, Massachusetts  Institute  of  Technology.  viii  + 
183  pp.     8vo.     $2.50. 

This  book  is  designed  primarily  for  classroom 
use  rather  than  for  reference  for  practicing  engi- 
neers. It  avoids  discussion  of  specialized  topics 
which  are  taught  separately  with  special  books  and 
devotes  itself  to  the  consideration  of  the  more  com- 
mon and  important  subjects.  At  the  end  of  each 
chapter  are  given  problems  to  illustrate  the  appli- 
cation of  the  principles  just  preceding. 

Benjamin's  Machine  Design. 

By  Charles  H.  Benjamin,  Professor  in  Purdue  Uni- 
versity.    i2mo.     202  pp.     $2.00. 

Machinery  : — We  know  of  no  v^^ork  on  Machine  Design 
which  can  be  more  heartily  recommended  to  the  average 
student  than  this.  .  .  .  The  work  has  the  characteristics  of 
Professor  Benjamin's  writing  in  general ;  that  is,  clearness 
and  simplicity.  It  is  brought  up  to  date,  containing,  for 
example,  a  summary  of  the  paper  on  the  collapsing  strength 
of  lap-welded  steel  tubes  presented  by  Professor  Stewart 
before  the  spring  meeting  of  the  A.  S.  M.  E.  in  1906. 

Leffler's  The  Elastic  Arch. 

With  special  reference  to  the  Reinforced  Concrete  Arch. 
By  Burton  R.  Leffler,  Engineer  of  Bridges  on  the  Lake 
Shore  and  Michigan  Southern  Railway.  viii-}-57  pp.  i2mo. 
$1.00. 

HENRY      HOLT      AND      COMPANY 

NEW  YORK  CHICAGO 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


290ct59lii 
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PCTisigsg 


8JuI'6UM 


RECn  1  D 


JUN  841961 


FEB    7  1969  3  8 
RECEIVED — 


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LOAN  DEPT. 


MAR  6     1975  4 


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LD  21A-50m-4,'59 
(A1724sl0)476B 


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